How to Make It
In a food processor fitted with the coarse grating disk, grate the carrots, onion, and zucchini. Heat the oil in a large pot over medium heat. Add the grated vegetables, 2 cloves of the garlic, 1 teaspoon salt, and ½ teaspoon pepper and cook, stirring occasionally, until beginning to brown, 12 to 15 minutes.
Add the crushed and whole tomatoes (with their juices) and oregano to the pot and simmer, stirring occasionally and breaking up the tomatoes with a spoon, until the sauce has thickened, 35 to 40 minutes.
Meanwhile, heat broiler. In a large bowl, gently combine the beef, bread crumbs, eggs, pecorino, the remaining garlic, and ½ teaspoon each salt and pepper. Form into 32 meatballs (about 2 tablespoons each) and place on a foil-lined large broilerproof baking sheet. Broil, turning once, until cooked through, 8 to 10 minutes. Add to the sauce and simmer for 5 minutes.
Cook the pasta according to the package directions. Top with the meatballs, sauce, and additional pecorino and serve with the salad.
The meatballs and sauce can be frozen together in freezer-safe containers for up to 3 months. To reheat, run the containers under warm water until the mixture slides out. Transfer to a saucepan and cook, covered, over low heat, stirring occasionally, until heated through. Cook the pasta and serve with the meatballs and sauce.