Heat the oven on the broil setting. Arrange the pita breads on a cookie sheet. Place the cookie sheet under the broiler until the pitas are toasted on one side, about 1-2 minutes. Flip the pitas and broil on the other side, another 1-2 minutes. Remove the pitas from the oven.
Spread each toasted pita with 2 tablespoons of pizza sauce. Top with chicken and sprinkle with mozzarella cheese.
Place the pita pizzas back under the broiler. Broil for 2 minutes or until hot and the cheese is golden and bubbly. Remove them from the oven and cut into wedges.
Serve warm with Parmigiano-Reggiano, crushed red pepper flakes, and dried oregano, if desired.