In a small bowl, soak the bread in the milk until the milk is absorbed, about 5 minutes. In a large bowl, gently mix the ground meats with the soaked bread, 2 tablespoons Parmigiano-Reggiano, 1/2 tablespoon of the garlic, 2 teaspoons salt, 1/4 teaspoon pepper, 1 tablespoon of the chopped basil, and the oregano. Form the meat mixture into golf ball-size meatballs and transfer to a baking sheet. Cover and refrigerate until chilled, about 30 minutes.

Meanwhile, in a food processor, pulse the tomatoes with their juices until finely chopped. In a large saucepan, heat 2 tablespoons of the olive oil. Add the remaining 1 tablespoon minced garlic and cook over moderate heat until light golden, about 3 minutes. Add the crushed red pepper, chopped tomatoes, and the basil sprig. Bring to a simmer and cook over moderately low heat until the sauce is slightly thickened, about 25 minutes. Discard the basil sprig and season the sauce with salt and pepper.

In a large skillet, heat the remaining 2 tablespoons of olive oil. Brown the meatballs in batches, about 3 minutes per side. Transfer the meatballs to the tomato sauce and simmer until cooked through, about 10 minutes. Discard the fat from the skillet. Add 1/4 cup of water and cook over moderately high heat, scraping up the brown bits from the bottom of the pan, about 2 minutes. Pour the water into the tomato sauce.

In a large pot of salted boiling water, cook the spaghetti until al dente. Drain and return the pasta to the pot. Pour in 2 cups of the tomato sauce and toss until the pasta is well coated, about 1 minute. Transfer the pasta to large shallow bowls. Spoon the meatballs and remaining tomato sauce over the spaghetti. Sprinkle with the remaining 2 tablespoons chopped basil. Serve at once, passing grated cheese at the table.

The meatballs and sauce can be cooked up to 2 days in advance. Reheat gently before serving.