Heat oil in large saucepan over medium-high heat until hot. Add onions; cook and stir 2 to 3 minutes or until tender. Add garlic and mushrooms; cook and stir 2 minutes.
Add broth; bring to a boil. Add tortellini; return to a boil. Boil 5 to 7 minutes or until tortellini are of desired doneness.
Stir in spinach; cook 1 to 2 minutes or just until wilted. To serve, spoon soup into individual soup bowls. Top each serving with cheese.