Stir bread crumbs, 3 tablespoons Parmesan cheese, 2 teaspoons oregano, salt, and black pepper together in a bowl.
Place tofu in a bowl of cold water. Place mushrooms in a separate bowl of cold water. Remove 1 mushroom from water; press mushroom into bread crumb mixture and turn to coat all sides. Transfer breaded mushroom to a plate. Repeat with remaining mushrooms and tofu slices.
Heat 2 tablespoons olive oil in a skillet over medium heat. Cook breaded tofu and mushrooms in hot oil until browned on one side, 2 to 3 minutes. Drizzle a little olive oil over the top, turn mushrooms and tofu slices, and cook until browned on the other side, about 2 minutes more.
Whisk tomato sauce, garlic, remaining oregano, and basil together in a bowl until sauce is smooth. Spread a thin layer of sauce into the bottom of an 8-inch square baking dish. Arrange tofu slices in a single layer over sauce, and top each tofu slice with a mushroom. Spoon remaining sauce over mushrooms and tofu; top with mozzarella cheese and remaining Parmesan cheese.
Bake in the preheated oven until cheese melts and mixture is bubbling, about 20 minutes.