1. Heat oil in a large pot over medium heat until hot but not smoking. Add sausage; cook, stirring with a wooden spoon to break up meat, until browned and crumbly, about 5 minutes. Add onion; cook, stirring often, until softened, about 5 minutes.

2. Add 4 cups water and the broth to pot. Bring to a boil. Stir in pasta. Reduce heat to medium-low, and simmer, stirring often, until pasta is barely tender, about 6 minutes. Add broccoli, tomatoes, and salt. Simmer until broccoli is tender, about 4 minutes. Ladle soup into shallow bowls, and top with cheese.