Heat a medium soup pot over medium heat. Add 3 tablespoons extra-virgin olive oil, garlic and 3/4 of the onion. Cook 7 to 8 minutes, then add tomatoes and crushed tomatoes and season with salt and pepper. Add stock and raise heat to make the soup bubble. Reduce heat to simmer and add bread and beans. Stir soup as it simmers until it thickens to a stew-like consistency. Turn off heat, adjust seasonings and ladle into shallow bowls. Top with grated cheese, an additional drizzle of extra-virgin olive oil and a spoonful of reserved finely chopped raw onions. Torn basil is an optional garnish.
Recipe courtesy of Rachael Ray