Rinse the rice and drain it well. Combine it with the ground beef, egg, minced onion, garlic, cilantro, mint, salt, and pepper, mixing only until combined. Shape the mixture into approximately 1 1/2-inch meatballs, making about 24–28, and chill them while preparing the soup.

Heat the vegetable oil in a large soup pot over medium heat. Sauté the chopped onion for 3–4 minutes, add the garlic, and cook for about 30 seconds.

Blend the tomatoes with about 1 cup of the broth until smooth. Pour the tomato mixture into the pot and cook for 5–7 minutes, stirring occasionally, until slightly thickened.

Add the remaining broth, carrots, potatoes, celery, bay leaves, oregano, salt, and black pepper. Bring the soup to a gentle boil.

Carefully add the raw meatballs one at a time. Lower the heat immediately so the liquid maintains a gentle simmer instead of a rolling boil.

Partially cover the pot and simmer for 25–30 minutes. The rice should be tender, the vegetables nearly cooked, and the meatballs should reach at least 160°F (71°C) internally.

Add the zucchini during the final 10–12 minutes of cooking. This timing keeps the zucchini tender and flavorful without allowing it to become mushy.

Remove the bay leaves, taste the broth, and adjust the salt if needed. Stir in the chopped cilantro and serve each bowl with lime wedges.