Combine garlic, tomato paste and stock in small pot, and stir to warm and loosen tomato paste. Add vinegar, brown sugar, oregano, red pepper and tomato sauce, and bring sauce to a bubble. Add basil and reduce heat to low simmer, simmer to thicken, 20 minutes.
Soak torn and crumbled bread in milk to soften. Place the ground meats in bowl. Squeeze excess milk from bread, add the crumbs to meat and season with salt and pepper. Add parsley, garlic, onion or granulated onion, and grated cheese; mix to combine.
Form 4 large patties making them thinner at the center for even cooking. Drizzle hot grill or griddle over medium-high heat with olive oil, cook patties about 8-9 minutes turning once and remove.
Brush 1 side of sliced bread with garlic butter. Build 4 sandwiches with buttered sides facing out layering bread, ketchup, mozzarella cheese, patty, provolone cheese, ketchup and bread. Griddle or grill until deeply golden and melted. Pass any extra ketchup at table.