1. Sort beans and discard debris. Rinse and drain beans.
2. Finely chop prosciutto.
3. In a 5- to 6-quart pan over high heat, stir carrot, celery, onion, and garlic in oil until onion is limp, about 4 minutes. Add prosciutto and stir just until mixture is lightly browned, 2 to 3 minutes.
4. Add beans, rosemary, 6 cups broth, 2 cups water, and tomatoes; bring to a boil. Cover, reduce heat, and simmer until beans are tender to bite, 2 to 2 1/2 hours; stir occasionally.
5. Purée 1 cup beans in a food processor or blender, then stir into beans. Add broth if beans are too thick. If making ahead, cool, cover, and chill up to 1 day. Reheat, thinning with more broth. Add salt to taste.