Cook onions and carrots in oil in an 8-quart pot over moderately low heat, stirring occasionally, until softened, 4 to 5 minutes. Add sausage and cook until it is no longer pink. Add garlic and cook, stirring, 1 minute. Add beans, broth, salt, pepper, bay leaf, and rosemary and simmer, uncovered, until carrots are tender.
Stir in kale and cook until it is your desired texture. Adjust seasoning to taste. Serve and garnish each bowl with Parmigiano Reggiano.