Preheat the broiler. Heat a large skillet over medium-high heat. Add 1 tablespoon olive oil and the mushrooms and season with salt and pepper. Cook, stirring occasionally, until the mushrooms are lightly browned, about 5 minutes; set aside.
Combine 1/4 cup olive oil and the garlic in a small bowl. Put the bread cut-side up on a baking sheet and drizzle with the garlic oil; season with salt. Broil until lightly toasted, 2 to 3 minutes. Spoon the ricotta on the bread, then top with the mushrooms and mozzarella. Broil until browned and bubbling, 2 to 5 minutes; transfer to a cutting board.
Cut the bread into pieces. Sprinkle with the parsley, parmesan and red pepper flakes. Drizzle the tomatoes with the vinegar and remaining 1 tablespoon olive oil and season with salt and pepper. Serve with the pizza.