Butterfly chicken breasts; cut in half. Pound 1/4 inch thick. Season with salt and pepper. In 3qt saute pan over medium heat, warm 1 cup oil to 350 degrees. Brush rolls on both sides with oil. Preheat panini press to medium.
Pout flour and eggs in separate breading trays. In another tray, combine crumbs and Parmigiano-Reggiano. Dredge chicken in flour; shake off excess. Dip into egg; let excess drip off. Place chicken in crumb mixture; press to adhere.
Fry chicken, 2 pieces at a time in saute pan, turning once, 3 minutes total. Drain on paper towels. Place 1 piece of chicken on bottom of each roll; top with 1/3 cup mozzarella, 1 Tbsp basil and 1/4 cup tomato sauce. Cover with top halves of rolls.
Cook sandwiches 2 at a time on panini press until cheese melts, 3-4 minutes.