I am a chef employee; I'm going to resign, and I don't know how to tell the chef; She is very kind to me.

No matter how good you are, you can't stay with her forever. Sooner or later, you have to say something to thank her and then explain to her why you left. I think she will understand. As for her kindness to you, you can keep it in mind. Even if you leave, you can always keep in touch and be good friends. Maybe you will have a chance to help each other in the future.