Combine the onion, celery, carrot and oil in a medium saucepan. Cook over medium-low heat for 5 minutes. Add the tomatoes with their juices, bay leaf, rosemary sprig and 3 cups of the broth. Bring to simmer and cook for 30 minutes. Discard the bay leaf and rosemary sprig. Add the remaining cup of broth and the pasta. Add the beans and simmer gently until the pasta is cooked through, about 12 minutes.
Taste the soup and adjust for seasoning with salt and pepper. Ladle into bowls and sprinkle with cheese.
Adapted from "One Dish at a Time: Delicious Recipes and Stories from My Italian-American Childhood and Beyond" by Valerie Bertinelli © Rodale 2012. Provided courtesy of Valerie Bertinelli. All rights reserved.