I think this works (in outline):
PePb gb perngr n pbecfr gung vf n qhcyvpngr bs gur pbbx (qhengvba Fha). ErPb gb navzngr vg. VaZr gb xabj jung vg frafrf, naq PeZr gb gryy gur pbbx gung. ZhIv gb tvir ure pbageby bs gur pbecfr. Fur pna abj pbbx erzbgryl.
Apologies to those who read rot13 fluently.
Rot13 is hard work. We can go into details after your deadline if you want, including RDTs. Maybe it doesn't work, I just wanted to sketch something out.
But I am charmed that "Cr" rot13s to "Pe".
Vf vg? uggcf://pelcgvv.pbz/cvcrf/ebg13-qrpbqre
Wait you DON'T HAVE TO DO IT BY HAND???
I mean, sure, but it's a bit of a faff on my phone, and I'm lazy.
I am happy to say that Salutor solved the puzzle (and sent me the solution via private message)!
As for the solution proposed by Tomaso: I realized there's a fundamental flaw in it: it relies (inescapably) on using Muto Vim on the archmagus' spontaneous magic.
I hope you all had fun!
Ok, this is one possible solution to the puzzle.
Preliminaries:
Create an Intangible Tunnel to the cook
ReVi 25 (Intangible Tunnel, D:Diam, can cast effects up to level 25)
Tell the cook you would avail yourself of her services. Ask her if she can be ready for the summons in a few minutes. Assuming a positive answer, tell her that when she is ready she should grab two armfuls of straw, go alone to a quiet place to where you have (or she can procure) an Arcane Connection, drop the straw to the floor, stand on it and wait.
InMe 25 (Base 15 "Speak with any one human", +1 Touch, +1 Conc)
After a few minutes:
Create in front of you an armful of straw
CrHe 4 ("Base 2 Create a processed plant product, like a finished plank of wood.", +1 Touch, +1 Diam)
Create an Intangible Tunnel to the cook
ReVi 25 (Intangible Tunnel, D:Diam, can cast effects up to level 25)
Get a view of what's going on, and make sure everything seems in good order
InCo(Im) 15 (Eye of the sage, R:Touch, D:Diam)
Ask the cook if she's ready, and tell her to relax
InMe 25 (Base 15 "Speak with any one human", +1 Touch, +1 Conc)
Make sure the cook won't fall even when she falls asleep.
ReCo 20 (Lifting the Dangling Puppet, R:Touch, D:Sun).
Put the cook to sleep (so she won't get startled by the experience and/or move around)
ReMe 5 (Call to slumber, but with R:Touch)
Turn the sleeping cook into a sleeping kitten
MuCo(An) 20 (Base 10 "Transform the target into a land animal", +1 Touch, +1 Diam)
Remember to add requisites for what she's wearing 
Turn the sleeping kitten into a kitten made of clay
MuAn(Te) 25 (Base 15 "Change an animal into a non-living item", +1 Touch + 1 Diam)
Turn the clay kitten into a little kitten-shaped cloud of blue air
MuTe(Au) 20 (Base 10 "Change dirt into a slightly unnatural liquid or gas", +1 Touch + 1 Diam).
Turn the kitten-cloud into a little fire, that burns the straw and nothing else (Rego requisite)
MuAu(Re,Ig) 15 (Base 4 "Transform an amount of air into another element (fire, earth, or water)", +1 Touch, +1 Diam + 1 Rego).
This is where the straw comes into play. A fire needs fuel, or it will go out. In principle, you could use just D:Mom, as you only need the cook in fire-form for the next round. Also, you could keep the fire in kitten shape adding one ("slightly unnatural") or at most two magnitudes ("very unnatural")
Summon the fiery cook to the straw in front of your feet
ReIg 25 (Base 3, "Instantly transport a fire up to 5 paces, +5 extend range to anywhere you have an Arcane Connection to (see TME, p.107), +1 Touch)
Turn the fire back into the kitten-cloud (you never know with fire ...)
PeVi 25 (Unravel the Fabric of Ignem, R:Touch)
Let the remaining shapeshifting spells expire on their own over the next minute or so.
As soon as the cook has resumed her human form, wake her up
ReMe 10 (Snap of Awakening)
And finally dispel the ReCo that was holding her
PeVi 25 (Unravel the Fabric of Corpus)
Essentially the same steps allow you to send the cook back.
The two key observations to solve the puzzle were:
a) it's much easier to instantly transport a fire than a person.
b) it's much easier to change a human into a fire by changing a human into other stuff, and other stuff into a fire. This general principle (it might be easier to change X to Y and Y to Z rather than X directly to Z) is crucial to playing a Hermetic shapeshifter to the hilt.
This solution contains a few unneeded steps:
I highlighted that the one presented was one possible solution!
You have a tradeoff between solving with fewer steps or with fewer books
But I'd spend a word about objects turned into fire burning without fuel. I've always found that statement both iffy, and fundamentally contradictory with what's written in TME, p.112 about The Chariot of Zeus, which transforms the caster into a lightning bolt. The text there says that "The Duration of this spell cannot be extended beyond Momentary, since bolts of lightning exist only for a moment". So it seems to say that, if you transform X into Y, you must still satisfy the natural constraints of Y. After all, if I transform a person into a fish, the fish would suffocate out of water even if a person would be able to breathe normally, right?
Isn't the simplest way to do this just to cast a level 15 tunnel on her, then a touch range inco base 10 spell, then just teleport her to you with a spell designed not to warp her through a similar large tunnel
Perhaps you need something like a nativity horoscope for her, but if you plan on doing this a lot you can probably find one day somewhere, or just get someone else to do it for you
I think, in the case of Chariot of Zeus, it has more to do with the nature of lightning which is ephemeral. Muto requisit would change this nature and make it last longer. As of Fire being able to sustain without fuel, that's an interesting one.
By analogy, human sustain on food and water, but can go without food and water for a few days - although their health will quickly deteriorate. By stretching this analogy to fire, could it be conceivable that fire create by magic could survive a few minutes, especially if this fire is coming from an initial target that can sustain ? Would there be a kind of loose connection between the essence of the initial target (the cook) that would prevent it to simply dying if turn into fire without fuel ? After all, fire elemental sustains without fuel.
Those are just hypothesis I am proposing to justify the existence of fire without fuel, compare to lightning.
Sure. But a normal fire can go for an instant (let's be generous: a round) without fuel before it dies out - see the ReIg examples in the corebook.
At the very minimum it would be an unnatural fire. I would accept that, but I think not every troupe might.
I think that the argument about the initial target is completely unfounded, however. Otherwise, a human transformed into a fish would not suffocate out of water (and may well drown in it); and a rock transformed into a horse, say for D:Moon, would never need to eat or drink or rest. In fact, the argument contradicts another example in MoH itself - the spell where a magus transforms his grogs into vermin so they can survive with less to eat.
Well, a fire elemental is not exactly the same thing as a fire. But more importantly, RoP:M p.139 says:
A fire elemental is often short-lived, lasting as long as it has fuel
The effect level for Hermetic teleportation is based on the distance the target is transported. Then Range is added on top of that if you're casting on someone else. So that doesn't really work.
There are a few teleportation powers that work differently: members of House Verditius can summon enchanted casting tools to themselves at a lower level, and the Incantation/Effusion guidelines are based on the destination's distance from the caster instead of from the target.
A touch range arcane connection teleport is level 40. Works fine through an arcane tunnel
The issue is that it's not that easy to design the spell not to warp her:
Pg 168 of the core rules
Anyone subjected to a powerful mystical effect gains a Warping Point, unless they themselves were responsible for the effect or it was specifically and carefully designed to work on them. “Powerful effect” is subjective, but any Hermetic spell of sixth magnitude or higher counts. Designing an effect for a particular target requires, in Hermetic terms, that a special version of a Formulaic spell be invented.
Over the longer term, the archmage can certainly design a formulaic that won't warp her, but that doesn't help him right now when he has to use spontaenous magic.
Yes. The puzzle was figuring out how to do it using spontaneous effects without warping the cook. Unless you come up with a way in which to design a formulaic spell in an evening, a spell designed for the cook is unavailable.
But you could potentially do this with spontaneous MuVi (to adjust the spell to this person) along with the standard ReCo effect you've used in the past, right? I'm not saying it works for all troupes, but it is potentially possible.
Ah that's fair, I didn't remember that it said that bit about formulaic magic.
Regarding what callen said, I would argue(ysmv) that not warping a particular subject is absolutely no more complex than adding +1 magnitude, which, at level 40, makes it doable by our Uber-archmagus and a bare handful of others in the order at most. Tbh i would probably argue that this is a superficial change of a spell(yeilding a much more atainable level 20), because it does not actually change the effect of the spell in any way, just the side effects.
Of note is that this effect(the warp-cusomization one) is not especially useful unless you can spont it, which requires at least a casting score of ~36 in muvi, relegating this to the realm of at least secondary specialists, the most arch of archmagi aside
I get the logic, however, I'd think preventing warping is one of those things that hermetic magic does very poorly. Conceptually you are adding more magic so the magic doesn't affect them. A bit counter intuitive.
Maybe after a season of study of the individual, analysing their horoscope, blood and stool samples, etc. Basically really personally invasive stuff; then I would allow a MuVi or +1 mag spont not warp that specific person.
At most I would think a formulaic spell that would MuVi another spell so it would not warp one specific individual.
That being said in reference to this particular challenge, is warping calculated by the magnitude of the original spell or the changed spell? If I cast a spell that is not strong enough to warp the target and then use MuVi to make it more powerful does it warp the target of the original spell?
This is one of the things that's been significantly debated. I feel it runs against the spirit of the "Designing an effect for a particular target requires, in Hermetic terms, that a special version
of a Formulaic spell be invented". If you can (re)tailor a formulaic ReCo spell to a target using spontaneous MuVi, why can't you tailor a spontaneous ReCo spell to the target from the outset?
Note that I would allow a Formulaic MuVi spell to tailor any other formulaic spell to a specific target (chosen when the MuVi spell is designed), but basically you have to put a season into designing at least one formulaic spell specifically for that target.
In any case the puzzle clearly said that you had to do it with spontaneous magic, not a combination of formulaic and spontaneous magic.