In a large, deep skillet or saucepan, bring the broth to a boil, then salt it and lower to a simmer.
Meanwhile, in a bowl, combine the ground meat, eggs, carrots, cheese, bread crumbs, parsley and garlic paste, if using; season with pepper. Form into 1-inch balls.
Drop the meatballs into the broth and simmer until cooked through, about 5 minutes. Stir in the pasta and cook until al dente, 6 to 7 minutes. Let cool before serving to your pooch! Season the people's portions with salt and pepper and sprinkle with more parmigiano-reggiano.