Heat the oil in a large Dutch oven or other heavy-bottomed pot over medium heat. Add the onion, carrots, and celery and cook, stirring occasionally, until softened, 5–6 minutes. Add the garlic, rosemary, and bay leaf and cook another minute until fragrant. Add the crushed tomatoes and vegetable broth and raise the heat to bring it to a simmer.
Place about a quarter cup of beans in a bowl and add a little bit of the cooking liquid. Mash them together to form a paste and add it to the pot along with the rest of the whole beans. Cover the pot and simmer the soup for 20 minutes. Add the pasta and simmer another 10 minutes, uncovered. Stir in the greens and cook another 5 minutes until wilted. Stir in the cheese.
Taste the soup and season with a pinch of salt and pepper to taste. Ladle the soup into bowls and top with extra grated cheese and a drizzle of olive oil. Soup will thicken as it stands.
To add extra flavor to this soup, add the rind from a block of Parmigiano-Reggiano cheese. As the soup simmers away, it will be infused with an unbelievable umami kick! Save your Parmigiano-Reggiano rinds and store them in your freezer.