1. Remove sausage from casings by trimming tops and squeezing from ends. Heat tablespoon oil in a heavy 5-quart pot over moderately high heat. Add sausage and cook, stirring and breaking up clumps, until no longer pink, about 5 minutes.
2. Transfer with a slotted spoon to a bowl. Add 1 tablespoon oil, the onion, and rosemary to same pot; cook over moderate heat, stirring, until onion is softened, about 3 minutes. Add sausage, beans, escarole, broth, and tomatoes; simmer, stirring occasionally, 15 minutes.
3. Ladle soup into bowls, sprinkle with cheese and drizzle with remaining 2 tablespoons oil. Makes 8 servings.