all the food classification discourse is phenotype-based and no more sophisticated than ancient greek zoology.
"pizza is a flat bread with toppings baked on"?
let me guess: you define a man as a featherless biped, don't you plato?
no, we need the taxonomical model. food evolves and adapts and descends. it fills niches that cause it to diverge from its ancestors and converge with other, unrelated species. unlike most animals, food is capable of horizontal gene transfer, making taxonomical divisions less linear.
american pizza (Focaccius rotunda) is descended from an italian baked flatbread dish. deep dish pizza (F. r. patina) is a subspecies of american pizza which evolved in the american midwest. it is taxonomically unrelated to common species of casserole in the area (and casserole is not a taxonomic category but a polyphleytic grouping, much like fish and crabs)