Pour the Basic Red Sauce into a large, heavy pot and bring to a simmer over medium-high heat, stirring occasionally. Place all the remaining meatball ingredients in a large mixing bowl and stir well to combine.

Using your hands, divide the meatball mixture into 12 even portions and roll into smooth meatballs. Place meatballs on a plate as they are formed. When all the meatballs are shaped, gently place the balls into the simmering sauce 1 at a time. Simmer the sauce, uncovered, for at least 10 minutes before stirring. The meatballs should begin to float to the top when they are able to be stirred. Stir gently and continue to simmer for an additional 20 to 30 minutes, stirring occasionally to prevent the sauce from sticking to the bottom of the pot.

Using oven mitts or pot holders, remove the sauce from the heat and cover to keep warm. Preheat the oven to 450 degrees F.

Using a sharp knife, slice a thin layer off the top of each hoagie bun and, using your hands, remove some of the bready interior to form a hollowed-out submarine shape. Brush the inside of the hollowed out buns with the olive oil. Divide half of the grated mozzarella evenly between the hoagie buns and bake, uncovered, until the cheese is melted and buns are lightly golden, about 3 to 4 minutes. Remove from the oven and ladle some of the hot tomato sauce into the buns. Place 2 or 3 meatballs inside each bun and top with more sauce. Divide the remaining mozzarella cheese among the tops of the buns and sprinkle with the Parmesan. Return to the oven and bake until bubbly and hot throughout and the cheeses are golden brown, 4 to 6 minutes.