Heat the olive oil in a soup pot over medium heat. Add onion, carrot, and celery, and cook, stirring, until the onion turns a pale gold color.
Add the pork (ribs, ham bone, or chops) and brown well on all sides.
Add the tomatoes and their juice and cook about 10 minutes, stirring from time to time. If using cooked dried beans, cook the tomatoes about 20 minutes.
Add the beans and the broth. Keep the liquid at a steady simmer for 45 minutes to 1 hour. If the level drops too low, add more broth as needed.
Put a food mill over a bowl and pass most of the contents through it, then return the purée to the pot.
Check the thickness. If it is too dense, add broth or water. Bring to a steady boil and cook for about 10 minutes.
Add the pasta and cook until done. Turn off the heat, stir in the butter and grated Parmigiano, and let the soup settle about 10 minutes before serving.
