Squeeze sausages from casings into an 8- to 10-quart pan over high heat and stir often, breaking them apart with a spoon, until browned and crumbly, 8 to 10 minutes. Spoon out and discard all but 1 tablespoon fat from pan.
Add carrots, onion, and garlic; stir often until onion is limp, 5 to 7 minutes. Add broth, tomatoes (including juice), beans, and basil and bring to a boil.
Add pasta, reduce heat, and simmer, covered, stirring occasionally, until pasta is just tender to bite, about 10 minutes. Skim and discard fat. Stir in spinach and cook just until it is wilted, about 30 seconds. Add salt and pepper to taste. Serve soup from the pan, or pour into a tureen. Offer parmesan cheese to add to taste.