Rustic Meatball Soup

Name: Rustic Meatball Soup
Ingredients: 1 pound ground beef; 6 large garlic cloves, minced; 1 large onion, finely chopped (about 3 cups); 2 medium green bell peppers, finely chopped; 2 teaspoons dried oregano; 3/4 teaspoon ground cumin; Kosher salt; 1 large egg, lightly beaten; 1/2 cup stone-ground yellow cornmeal; 2 tablespoons peanut oil; 1 pound plum tomatoes, seeded and cut into 1/4-inch dice; 1 large carrot, cut into 1/4-inch dice; 2 quarts plus 2 cups water; 6 cilantro sprigs, plus 1 tablespoon chopped cilantro; 1/2 cup long-grain rice; Freshly ground pepper; Lime wedges, for serving