You certainly can Max Cooking Career, and well before your young adult reaches adult status.
Join cooking career on day one of your sim's life. I had ambitious, workaholic, bookworm, friendly and natural cook as traits. Obviously you will have to prepare some food during the sim's life and this will improve the cooking skill without book study. Set the work mod to "practice cooking" and change to "suck up to boss" if you need to speed up the promotion meter a bit). As soon as you have signed up for career go to the library and use the computer to "work from home". Shouldn't take long to reach about one third full on the work meter, this will ensure promotion next day. It will take ages to go through the whole process, but if you get your sim to half way mark on the meter before work, they should get next day promotion regardless of the cooking skill. I made sure to build relationship on the phone with boss first, then co-workers as soon as possible by taking a trip to France (the most comfortable lodgings) and spending my time chatting to them. Of course I could have used this time reading about cooking or practicing cooking instead of chatting, but chatting is more fun for the sim. Don't forget to buy the recipes of France before returning home if you want menu maven as I did. Do all work challenges offered if you can. The cooking and gardening ones are unlikely to be possible.
My sim had a very red face for cooking when she walked out of the resturaunt with the super fridge (the best job reward in the game, in my opinion, as food never spoiles). I don't remember what her cooking skill was, but certainly no more than 5 as she is only 6 now.
Level 10 was the only level that took 2 days for my sim.
I quit immediately and joined the music career! Much better for making friends as I can put in an hour or so after work busking.