Heat oven to 425°F and set a wire rack over a rimmed sheet pan. Pound each chicken cutlet to an even 1/4 inch, then season with 1/2 teaspoon salt and pepper.
Put one mozzarella stick near a short edge of each cutlet, fold the sides inward and roll tightly around the cheese. Secure each with two toothpicks, keeping track of all eight.
Put flour in one shallow dish, beat eggs in another and combine panko with 1/3 cup Parmesan in a third. Coat each chicken roll in flour, egg and crumbs, then place seam side down on the rack and brush with 1 tablespoon oil.
Bake for 25 to 35 minutes until the center of each roll, including the filling area, reaches 165°F. Rest for 5 minutes, remove all eight toothpicks and slice each roll into four thick rounds.
Meanwhile, heat remaining oil in a saucepan, cook garlic for 30 seconds and add tomatoes, oregano and remaining salt. Simmer for 15 minutes until thick.
Toast the cut roll faces under the broiler for 1 to 2 minutes. Arrange the four chicken rounds from one roll on each bottom bun, spoon tomato sauce over the centers and sprinkle with the remaining Parmesan.
Close with the tops and serve one sandwich per person while the cheese centers are still soft.
