Place flour, eggs and breadcrumbs into three large, shallow containers or bowls. Season the chicken generously with salt and pepper. Dip a piece in flour, then eggs, then coat with breadcrumbs. Repeat until all of the chicken is coated.
Fill a large skillet with 1/2-inch oil. Place over medium-high heat, and when the oil is hot, fry cutlets in batches, turning halfway through, until golden brown. Transfer to a paper towel-lined plate.
Spoon a thin layer of sauce over the bottom of a 9-by-13-inch baking pan. Sprinkle one-third of the parmesan over sauce. Place half of the cutlets over the parmesan and top with half the mozzarella pieces. Top with half the remaining sauce, sprinkle with another third of the parmesan, and repeat layering, ending with a final layer of sauce and Parmesan.
Bake for about 40 minutes until the cheese is golden brown and the casserole is bubbling. Let cool a bit before serving.