If I understand your explanations correctly, I suppose that one could say about the famous American cookbook author
Julia Child: Она была
ученицой французской кухни. (Because she "threw herself", body and soul, into learning the methods of great French chefs, and really made herself a "disciple" of
haute cuisine -- and then wrote cookbooks and had a TV cooking program to show Americans how to cook French-style.)
However, for someone like me, who is maybe interested in learning a few different French recipes, but has no interest in studying ONLY French cuisine, to the exclusion of Italian and Chinese and Middle Eastern cooking styles, it's not appropriate to say
ученик французской кухни, because I'm not a "disciple" of the "art of French cooking."