In a large bowl, combine the milk and bread crumbs. Add the ground pork, veal, beef, the 1/3 cup parsley, the eggs, garlic, salt and pepper. Mix briefly with your hands. Form the mixture into 2-inch balls. Press a mozzarella cheese cube into the center of each ball, sealing it inside.
In a large skillet set on medium-high heat, heat 1/2 inch of oil until almost smoking. Add the meatballs and cook until browned, about 1 minute per side. Transfer to a paper towel-lined tray.
Discard the oil in the pan. Pour the tomato sauce into the pan and bring to a simmer over medium-high heat. Add the meatballs, reduce the heat to medium-low, cover and simmer until cooked through, about 30 minutes. Uncover and cook for 10 minutes more.
Put the pasta in a warmed large, shallow bowl. Top with the sauce and meatballs and garnish with parsley. Serve immediately and pass the Parmigiano-Reggiano cheese at the table.
Tomato Sauce
Place the tomatoes in a tomato press or food mill set over a bowl and pass them through the press or mill. Repeat as needed until all the large pieces of tomatoes are crushed. Set aside.
In a large saucepan over medium heat, warm the olive oil. Add the onion, salt, pepper and chili flakes, and cook, stirring frequently, until the onion is tender and translucent, about 10 minutes. Add the garlic and cook, stirring frequently, for about 2 minutes; do not allow the garlic to burn. Add the oregano and stir to combine. Stir in the crushed tomatoes. Bring to a boil, reduce the heat to medium-low and cook, uncovered, until the sauce is thickened, about 30 minutes. Stir in the basil and cook for 1 minute.
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