Preheat oven to 350°. Prepare breading station with 3 large mixing bowls: one with bread crumbs and Parmesan mixed together, another with eggs beaten with 1 tablespoon water, and the last with flour. Season chicken with salt and pepper. Coat each piece of chicken in flour and shake off excess, then dip into egg and coat in bread crumbs. Set chicken aside on a plate.
In a large cast-iron skillet over medium-high heat add 1" oil. When pan is hot but not smoking add chicken cutlets. Cook until golden brown, 1 to 2 minutes on each side. Transfer to a paper towel-lined plate.
In a small sauce pan, warm up tomato sauce.
Meanwhile, in a rimmed sheet pan lined with parchment paper, place hero rolls cut-side up. Toast until lightly golden brown, 1 to 2 minutes. Place chicken cutlets on bottom halves of hero rolls and top with mozzarella. Toast again until cheese begins to melt. Add spoonfuls of warm tomato sauce and 3 or 4 fresh basil leaves between the chicken and hero rolls. Close sandwich and cut in half. Serve immediately.