Pour the olive oil into a large saucepot and place over medium heat. Add the onions, garlic, carrots, and basil and saute for 10-12 minutes, stirring regularly.
Open the cans of stewed tomatoes. Pour the juices into the pot, then squash each tomato by hand, breaking into irregular pieces, as you place them into the pot. Add the bay leaf, 1 teaspoon salt, 1/2 teaspoon ground pepper, vegetable broth, wine, and bread. Bring to a boil, lower the heat, and simmer for 20-30 minutes. Stir occasionally.
Check the pappa al pomodoro. If the bread cubes haven't dissolved creating a thick porridge, use a potato masher to break them up into smaller pieces. Then turn off the heat and stir in 3/4 cup shaved Parmesan cheese.
To serve, scoop the soup into cups or bowls and finish with a little drizzle of olive oil and a sprinkle of Parmesan cheese.