Heat oven to 375°F. Cook mushrooms and onion in oil in wide saucepan 10 to 12 minutes until water evaporates. Add garlic 30 seconds.
Add tomatoes, sauce, basil, sauce salt, and pepper; simmer 10 minutes. Meanwhile, boil spaghetti with tablespoon salt to 3 minutes short of package time; drain.
Thoroughly tong-toss spaghetti with sauce to coat long strands. Transfer to deep 9-by-13-inch dish.
Scatter mozzarella and Parmesan evenly; foil-cover and bake 20 minutes.
Uncover; bake 8 to 10 minutes until center 165°F and top lightly patch-browned. Rest 10 minutes; lift with tongs and wide spoon.