From the first part of the sentence we know that the recipes are modified, so the meals based on them would change too. For the first blank, we need a word that implies change. They must be changed through some sort of action - this action might be connected to the different styles that we know those cooks represent. The word describing this action fits the second blank.
Out of the possible answers, "obsolete" in (A) means "no longer used," which is indeed connected to change - but we're talking about popular dishes, so they're still being used. "Enriched" in (C) is a better word here, as it means "improved by adding something." The second word in this answer is "innovation," which means "doing something in a new way"—this "new way" might come from different styles of cooking, so it fits the second blank perfectly.
The correct answer is (C).