Cold soak or quick soak the beans: Cold soak-dump beans into a 2-3 quart container; pour in enough water to cover by at least 4 inches; let soak in a cool place at least 8 hours or up to 24 hours; drain thoroughly. Quick soak-place beans in a large saucepan, covering them generously with cold water; bring to a boil; boil 1 minute before removing from heat; let stand uncovered 1 hour; then drain.
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 sauteed vegetables, about 25 minutes.
Process the bacon and garlic to a paste in the food processor, stopping once or twice to scrape down the sides of the work bowl.
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 onion and cook, stirring, until translucent, about 4 minutes.
Stir in the carrots and cook until the onion begins 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 after adding the vegetables from the skillet.
Ladle about 1/3 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 they are no longer steaming; process the beans and liquid in a food processor or blender until creamy; return the pureed beans to the pot.
Fish out the potatoes onto 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.
Let the soup rest off the heat, covered, 10-15 minutes.
While the soup is resting, cook the ditalini in salted water until very al dente (cook all the pasta if serving the full recipe of soup, or a proportionate amount if you are setting some of the soup aside for later).
Drain thoroughly and stir into the soup; let all rest 5 minutes, then serve in warm soup bowls, with a drizzle of oil and a sprinkle of Parmigiano-Reggiano.