Quick-soak beans by placing them in a large saucepan, covered with cold water and bringing them to a boil. Boil 1 minute. Remove from heat and let stand uncovered 1 hour, then drain and continue with the recipe.
Pour 4 quarts of water and 6 cups of chicken stock into a large stockpot. Add the drained beans, potatoes, rosemary and bay leaves. Bring to a boil over high heat and then adjust to a gentle boil. Let boil about 25 minutes.
Process the bacon and garlic to a paste in a food processor. Heat the oil in a large skillet over medium heat. Cook the bacon-garlic paste until golden, about 5 minutes. Stir in the onion and cook 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 the contents of the skillet into the soup pot. Season with salt and pepper and simmer until beans are tender, 45 minutes to an hour after adding the vegetables. When soup is almost ready, add ditalini pasta and cook until al dente. Let soup rest for about 5 minutes, then serve with a drizzle of extra-virgin olive oil and a sprinkle of Parmigiano-Reggiano.