Preheat oven to 375° F. Lightly coat a two-quart baking dish with oil or non-stick cooking spray; set aside. Heat oil in large pot or Dutch oven over medium heat. Add onion, garlic and carrot; sauté 10 minutes, stirring often. Stir in tomatoes with juice, breaking them up with a spoon. Add tomato sauce, mushrooms and oregano. Bring to a boil, then reduce heat and simmer 10 minutes. Stir in beans and heat through. Remove pan from heat.
Add drained pasta, shredded cheese; crumble in ricotta or tofu; toss gently. Spoon mixture into prepared baking dish and sprinkle with Parmesan cheese. Cover and bake for 30 minutes.