Heat oil in a medium saucepan over medium-high heat. Add carrot, onion, and garlic; sauté 5 minutes. Add the oregano, salt, pepper, broth, and tomatoes; bring to a boil. Cover, reduce heat, and simmer 10 minutes.
Increase heat to medium-high; add pasta, beans, and spinach; cook 14 minutes or until pasta is done, stirring occasionally. Sprinkle each serving with cheese.