Preheat oven to 475 degrees F. Line a large baking sheet with parchment paper; set aside. In a large bowl combine asparagus, sweet peppers, onions, mushrooms, oil, and salt. Spoon vegetable mixture onto prepared baking sheet, spreading into an even layer. Roast for 10 minutes; remove from oven and set aside.
Place flatbreads on two large baking sheets; sprinkle Pecorino-Romano cheese evenly over flatbreads. Arrange the roasted vegetables and the cherry tomatoes evenly on flatbreads. Sprinkle with mozzarella cheese. Bake about 5 minutes or until cheese is melted.