Heat the oven to 425°F. Toss eggplant and mushrooms with 2 tablespoons oil, 1/2 teaspoon salt and pepper on a large rimmed sheet. Roast for 25 to 30 minutes, turning once, until eggplant is tender and vegetables are browned.
Meanwhile, heat remaining oil in a soup pot over medium heat. Cook onion for 7 minutes, then stir in garlic and tomato paste for 1 minute.
Add crushed tomatoes, broth, oregano and remaining salt. Simmer uncovered for 15 minutes.
Stir roasted vegetables and fresh pasta ribbons into the soup. Simmer gently for 3 to 5 minutes, stirring carefully, until pasta is tender.
Mix ricotta and Parmesan-style cheese and divide among 6 bowls. Ladle hot soup over the cheese mixture and top each bowl with an equal share of mozzarella; let stand for 1 minute to soften the cheese before serving.