It does seem creating cooked food is possible due to proficiency in cooking tools
The base description of this spell is about using unworked (raw) materials and converting them into products of the same material.
It's unclear, and likely up to a DM, if that works with cooking. In the standard fabrication of products, you aren't necessarily chemically changing the properties of the original materials - you are just working them into something greater.
Having said that, there is the possibility of changing those properties. Working some materials that require craftsmanship (armor, weapons, jewelry) often does involving working the material in a way that chemically changes it. If you take that tack, then the spell can do this (but requires the proficiency that this Wizard has.)
But whether or not a DM wants to allow that type of change I think is up for table debate but the argument is very strong for allowing it.
How complicated is the dish?
Fabricate also explicitly states:
You convert raw materials into products of the same material...You can fabricate a Large or smaller object (contained within a 10-foot cube, or eight connected 5-foot cubes), given a sufficient quantity of raw material.
It is unclear if you can fabricate from multiple types of raw materials, but in all of the examples you are converting one raw material into one finished product. There isn't mixing and matching of various raw materials to create a new complex product.
Is it hot? A slippery slope for higher temperature finished products
There is also a concern about having this spell do more than what it says with regard to temperature of the end product. The fabrication process describes ending in mundane objects that are seemingly of standard room temperature. While the process of fabricating a sword mundanely requires high heat, the spell is simply skipping that step and producing the final usable product.
If you do allow fabricate to create final food products, then it should go with the rest of fabricated items and have them be at room temperature. Should cooked food come out at higher temperatures, then you could conceivably fabricate a sword whose blade is at high temp and thus delivers fire damage upon striking (if done quickly enough after fabrication.)
The issue of slippery slope is one to consider by the DM. Limiting it to food creation for Rule of Fun seems reasonable, but extrapolating that acceptance to mechanical advantages during combat might be where a DM draws the line.
Sure, but why?
I don't think it's unreasonable to allow it, and the proficiency in cooking tools that the wizard has solves the craftsmanship issue, but I'm not really sure what the purpose would be (other than not lighting a fire.)
Have everything you need?
In addition, in order to make a complete dish a DM could demand that you'd need all relevant ingredients (raw materials). You can't fabricate Beef Wellington with only raw meat. Even in the case of a simple stew...you need herbs and spices.
And for that, there are other ways to eat.
If this is ultimately about time and taste, then a combo of Create Food and Water and prestidigitation would provide you with flavorful food in a much faster timeframe with less resource expenditure (3rd level spell vs 4th level for fabricate. And you don't even need to find all the raw materials. This combo first create:
food is bland but nourishing
You chill, warm, or flavor up to 1 cubic foot of nonliving material for 1 hour.
to make it tasty and palatable.
The real option: Make a Heroes' Feast
You could also get what you're really looking for..and more with heroes' feast. It's a 6th level spell, but when it happens it'll be great for the party and the guest.
You bring forth a great feast, including magnificent food and drink. The feast takes 1 hour to consume and disappears at the end of that time, and the beneficial effects don't set in until this hour is over. Up to twelve creatures can partake of the feast.