Saute onion, garlic, and mushrooms in oil in a large saucepan until tender, 5-8 minutes. Stir in rice and cook 2-3 minutes.
Heat broth to boiling in medium sucepan; reduce heat to low. Add 1 cup broth to rice mixture nad cook, stirring constantly, until broth is absorbed, 1-2 minutes. Slowly add 2 cups broth and simmer, stirring until broth is absorbed, 8-10 minutes.
Add shrimp, snap peas, and remaining broth to saucepan. Cook, stirring frequently, until rice is just tender and liquid absorbed, 5-10 minutes. Add beans and tomatoes; cook 2-3 minutes longer. Stir in cheese; season to taste with salt and pepper.