Congratulations, you've progressed from kitchen assistant to cook. Most managers don't take ultimatums well, so you need to ask yourself what you really want. Getting a raise to cook where you are would be great, but is the raise part or the where you are part more important?
If it's where you are, ask now for a raise and if you get a yes, terrific. If you don't, oh well, at least you're cooking and you get to stay somewhere familiar.
If it's the raise part, ask now for a raise and if you get anything other than a yes (including "we'll see, maybe in a few months when business is better") make a sad face and say that you're disappointed. No threats. Then quietly go and look for a new job. When you get one, tell your boss that you're leaving and be nice about it.
Very few people work in the same kitchen all their professional lives. Moving jobs for more money, better hours, more responsibility and so on will be your normal until you retire. You might as well get used to it. In the future, it won't always be your choice (restaurants close, business slumps and people are laid off, and so on) so you need some practice at finding and landing kitchen jobs. Now is a fine time to start, while your educational institution may be able to help.