Prepare the cucuzza: Peel the cucuzza’s thin green skin and cut it into small cubes.
Sauté aromatics: In a large pot, heat olive oil over medium heat. Add onions and cook until translucent, about 5 minutes. Stir in garlic and cook 1 minute more.
Add tomatoes and cucuzza: Stir in chopped tomatoes and cucuzza pieces. Cook for 5–6 minutes, letting the flavors blend.
Simmer: Pour in broth and season with salt, pepper, and basil. Bring to a boil, then reduce heat to low. Cover and simmer for 25–30 minutes, until cucuzza is soft and tender.
Optional additions: For a heartier version, stir in cooked beans or small pasta during the last 10 minutes.
Serve: Ladle into bowls, drizzle with olive oil, and top with grated cheese if desired.