1 pound canellini (white kidney) beans
3 large Idaho potatoes (about 1 3/4 pounds), peeled
6 quarts water
3 sprigs fresh rosemary
2 bay leaves
12 slices bacon, cut crosswise into 1/2-inch strips (about 1 cup)
4 garlic cloves, peeled
¼ cup extra-virgin olive oil, plus more drizzling over the soup
1 medium onion, chopped (about 1 cup)
2 medium carrots, peeled and coarsely shredded (about 1 cup)
2 cups canned Italian plum tomatoes, preferably San Marzano, with their liquid, crushed
freshly ground black pepper
1 pound ditalini or 3 cups elbow pasta
Parmigiano-Reggiano, freshly grated
Cold-soak or quick-soak the beans. It's always a good idea to pick over dried beans to remove any dirt or tiny stones. Then dump the beans into a 2 to 3 quart container and pour in enough cold water to cover them by at least 4 inches. Let soak in a cool place at least 8 hours or up to 24 hours. Drain thoroughly.
Pour 6 quarts water into a tall, large (at least 10-quart) pot. Add the drained beans, potatoes, rosemary and bay leaves. Bring to a rolling boil over high heat, then adjust the heat to a gentle boil. Let boil while preparing the sautéed vegetables.
Process the bacon and garlic to a paste in a food processor, stopping once or twice to scrape down the sides of the workbowl. Heat the oil in a large skillet over medium heat. Scrape in the bacon-garlic paste and cook, stirring, until golden, about 5 minutes. Stir in the onions and cook, stirring, until translucent, about 4 minutes. Stir in the carrots and cook until the onions begin to brown, about 5 minutes. Add the crushed tomatoes, bring to a boil, then lower the heat and simmer for 5 minutes.
Pour two ladlefuls of the bean cooking water into the skillet and bring to a boil, then pour the contents of the skillet into the soup pot. Season lightly with salt and pepper and bring to a slow boil. Cook until the beans are tender, 45 minutes to 1 hour.
Ladle about one third of the beans along with enough cooking liquid to cover them into a baking dish or other shallow container where they will cool quickly. Cool the beans until no longer steaming. (Blending or processing hot beans can cause splatters. Stirring will speed up the cooling process.) Process the beans and liquid in a food processor or blender until creamy. Return the beans to the pot.
Fish out the potatoes to a plate. Mash them coarsely with a fork and return them to the pot. Cook 10 minutes to give the flavors a chance to blend. (If you're setting some of the soup aside to serve at another time, ladle it off now, before adding the pasta.) Let the soup rest off the heat, covered, 10 to 15 minutes.
While the soup is resting, cook the ditalini or elbow pasta in salted water until very al dente. (Cook all the pasta if serving the full recipe of soup or a proportionate amount if you're setting some of the soup aside for later). Drain thoroughly and stir into the soup. Let all rest for 5 minutes, then serve in warm soup bowls, with a drizzle of extra virgin olive oil and a sprinkle of Parmigiano-Reggiano.