Slice each chicken breast down the center to make 4 cutlets. Place each between 2 pieces of plastic wrap and pound to about 1/2 inch thinness with a meat mallet.

Marinate chicken breasts in Italian dressing in the refrigerator overnight.

Remove chicken to bring to room temperature 1 hour before cooking.

Combine flour and baking powder in one bowl, milk and egg in another bowl, and bread crumbs in a third bowl.

Heat 1 tablespoon butter in a pan over medium heat. Cook prosciutto in the hot butter until it starts to brown, about 5 minutes. Transfer to a paper towel-lined plate.

Bring marinara sauce to a simmer in a saucepan.

Heat oil in a deep-fryer or large saucepan to 375°F (190°C). Preheat the oven to 450°F (230°C).

Lay chicken cutlets out flat. Line 1/4 of mozzarella cheese down the center of each and sprinkle 1/4 of prosciutto over cheese. Fold cutlets in half over the stuffing and secure edges together with toothpicks.

Dunk rolled chicken into the flour mixture, then egg mixture, and finally bread crumbs.

Drop chicken into the preheated fryer and cook until crumbs are golden brown, chicken floats, and is no longer pink in the center, 6 to 7 minutes. Transfer to a paper towel-lined plate.

Cut sub rolls down the centers and spread remaining butter into interiors.

Assemble sandwiches: bottom sub half, chicken cutlet, marinara, provolone cheese, Parmesan cheese, and top sub half. Transfer subs to a baking pan.