In a large mixing bowl, combine the ground beef, uncooked rice, egg, chopped mint, minced garlic, cumin, salt, and pepper. Mix gently with your hands just until combined. Roll into golf-ball-sized spheres.
Heat the olive oil in a large soup pot or Dutch oven over Medium heat. Add the onion and cook for 3 minutes until soft. Add the garlic and chopped tomatoes. Cook for another 3–4 minutes until the tomatoes break down.
Pour in the beef broth. Bring the liquid to a rolling boil. Once boiling, reduce the heat to a simmer.
Carefully drop the meatballs one by one into the simmering broth. Let them poach for 5 minutes to firm up the exterior.
Add the carrots and potatoes. Cover the pot and simmer for 20 minutes.
Add the zucchini and simmer for just 5 more minutes. Taste the broth and adjust salt.
Ladle into bowls, ensuring everyone gets 3–4 meatballs and plenty of veggies.
