In a medium bowl, soak the bread crumbs in the milk.
In a large, wide saucepan, heat the olive oil over medium heat. Add the smashed garlic and cook until golden, about 2 minutes. Add the tomatoes, season to taste with salt and let the sauce simmer over low heat, stirring occasionally.
Add the beef, chopped garlic, parsley, Parmigiano-Reggiano, egg and 1 teaspoon of salt to the soaked bread crumbs and stir with a fork until combined. Place about 2 tablespoons of the mixture in your hand and press a mozzarella cube into the center. Shape the meat around the cheese, forming a ball. Repeat with the remaining meat and mozzarella.
Stir the sauce and raise the heat to medium- low. Carefully place the meatballs in the sauce, submerging them completely. Bring the sauce to a simmer and cook without stirring for 20 minutes.
Meanwhile, bring a large pot of salted water to a boil. Add the spaghetti, cook until al dente, about 8 minutes, and drain. Toss the spaghetti with the tomato sauce and meatballs, sprinkle with Parmigiano-Reggiano and serve.