Do not let players use some homebrew they find online. It’s almost always overpowered garbage.

I’d recommend that you talk to the player about what exactly is the fantasy they are looking for.

The D&D classes are primarily packages of mechanics for what a character does inside of combat. One does not cook during combat. Realistically, being a cook is more about how you roleplay outside of combat. They can pick whatever class they want, as long as they support the cook fantasy with their other choices.

You could maybe work with the player to come up with a custom background and have them use the “Chef” feat from Tasha’s Cauldron of Everything as the background bonus, which is only mechanical support you will find for being a cook in 5e.

If your player wants to be Senshi from Delicious in Dungeon, they are going to have a bad time. There are simply no mechanics to support that much cooking. Nutrition isn’t modelled in D&D, there isn’t a monster harvesting system, every table I’ve ever played at forgets to even mark off rations during rests. You could homebrew a whole new mechanic for food, but this would be a big effort and may be hard to balance in a way that’s fun for everyone at the table.

Don’t be afraid to tell the player “no” if you don’t think D&D is the right game to achieve the fantasy they want. It’s better to rip that Band-Aid off quickly rather than have the game system do it over several sessions.

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Re flavoring spells seems like the easy way to go about this:
Acid arrow: marinate & tenderize!
Any fire spell: flambe!
Viscous mockery: IT’S RAW!
Mage hand: sous chef.

Or, think about it differently, like Sanji from one piece, where cooking is his thing, but he’s also a solid fighter. (That will not use knives or his hands to fight, those are for cooking)

Ah the culinary artificer. It all started with banging rocks and sticks together. That’s how I discovered cinnamon and my life changed.

I’d suggest this http://dnd5e.wikidot.com/artificer:alchemist

And just give them double proficiency or advantage on anything to do with food.

Embrace the flavor. And by flavor, I mean change the theme of your character without changing the mechanics.

Is that a dagger or an incredibly sharp sushi knife? Is that a longsword or a butcher’s cleaver?

I personally have made this character as an artificer but you could do it as a rogue. A highly skilled chef who knows exactly how to cut (sneak attack). As an adventurer, you seek out new flavors and styles of cooking and aren’t affraid to try some dragon steaks, with harpy eggs, with a side of black ooze gelato.

Also there is a single feat that really helps push it.

I’d say to treat it like a profession and not a class. Like you would a Blacksmith or Barmaid.

Maybe there are some custom profession stuff available somewhere you can build off, but yeah, most of it would be flavor (pun intended).