Make the warm ketchup: In a small pot, combine the tomato paste, stock and garlic over low heat; stir to warm and loosen the tomato paste. Add the tomato sauce, vinegar, brown sugar, oregano and red pepper and bring the sauce to a bubble over medium heat, stirring often. Add the basil and reduce the heat to low; simmer to thicken, 20 minutes.
Make the burgers: Soak the crumbled bread in milk to soften. Place the ground meats in a bowl. Squeeze the excess milk from the bread, add the bread to the meat, and season with salt and pepper. Add the 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. Preheat a grill or griddle over medium-high heat; drizzle with olive oil and cook the patties, turning once, 8 to 9 minutes for medium-rare.
Brush 1 side of each bread slice with garlic butter. Build four sandwiches, buttered sides facing out, layering half the bread with warm ketchup, the mozzarella, patty, provolone, more warm ketchup and the remaining bread. Grill or griddle over medium heat, turning once, until the bread is deeply golden and the cheese is melted. Pass any extra ketchup at the table.