Continue cooking and lifting edges until egg mixture is set but is still glossy and moist. Sprinkle with half of the mozzarella cheese. Top with half of the mushroom mixture. Continue cooking until cheese just begins to melt. Using the spatula, lift and fold an edge of the omelet partially over filling. Transfer omelet to a warm plate. Prepare another omelet with remaining oil, egg mixture, mozzarella, and mushroom mixture. Top omelets with Asiago or Parmesan cheese, tomato, and, if desired, parsley.