grilling vs cookery

grilling and cookery both are nouns.

Word	Noun	Adjective	Verb	Adverb
grilling	Yes	No	No	No
cookery	Yes	No	No	No

As nouns, cookery is a hypernym of grilling; that is, cookery is a word with a broader meaning than grilling:

grilling: cooking by direct exposure to radiant heat (as over a fire or under a grill)
cookery: the act of preparing something (as food) by the application of heat

Other hypernyms of grilling include cooking, preparation.

grilling (noun)	cookery (noun)
cooking by direct exposure to radiant heat (as over a fire or under a grill)	the act of preparing something (as food) by the application of heat