Heat broiler. Heat 3 tablespoons of the oil in a large skillet over medium heat.
Season the chicken with ½ teaspoon salt and ¼ teaspoon pepper. Dredge in the flour, then in the eggs (letting any excess drip off), then in the bread crumbs.
Working in two batches (using the remaining 3 tablespoons oil for the second batch), cook the chicken until golden brown, 2 to 3 minutes per side. Transfer to a paper towel-lined plate.
Cut the cutlets in half crosswise. Place 4 halves on a baking sheet, spoon half the sauce over them, top with the remaining cutlets and sauce, and sprinkle with Parmesan.
Broil until the cheese melts, about 2 minutes. Divide among plates. Serve with the garlic bread (if desired).