Heat olive oil in a large pot over medium heat. Add ground beef and season lightly with salt and pepper. Cook, breaking it up, until browned. Drain excess fat and transfer beef to a plate.
In the same pot, add a drizzle of oil if the pot is dry and then add onion and bell peppers. Cook 3–4 minutes until softened. Stir in garlic and cook 30 seconds.
Add tomato paste and cook 1–2 minutes, stirring frequently, until slightly darkened.
Stir in diced tomatoes, tomato sauce, beef broth, Worcestershire sauce, bay leaf, parsley, oregano, basil, and red pepper flakes. Return cooked beef to the pot.
Bring to a light boil, then reduce heat to low. Cover and simmer 25–30 minutes, stirring occasionally.
While the soup simmers, cook rice according to package directions.
Remove and discard bay leaf. Taste and adjust salt and pepper as needed.
To serve, spoon cooked rice into individual bowls and ladle soup over the top. Garnish with mozzarella and additional parsley if desired. Serve warm.