- Butterfly the chicken breasts, cut each one in half and pound to 1/4-inch thickness. Season with salt and pepper.
- In a 3-quart sauté pan over medium heat, warm the 1 cup oil to 350°F. Brush the rolls on both sides with oil. Preheat an electric panini press to medium. Line a baking sheet with paper towels.
- Put the flour and eggs in separate breading trays or bowls. In another tray or bowl, stir together the bread crumbs and Parmigiano-Reggiano. Dredge the chicken in the flour, shaking off the excess. Dip the chicken into the eggs, letting the excess drip off. Place the chicken in the bread-crumb mixture, pressing gently so the crumbs adhere.
- Place 2 chicken pieces in the hot oil in the sauté pan and fry, turning once, until golden and crisp, about 3 minutes total. Transfer to the paper towel-lined baking sheet. Repeat to fry the remaining chicken.
- Place 1 piece of chicken on the bottom of each roll and top with 1/3 cup mozzarella, 1 Tbs. basil and 1/4 cup tomato sauce. Cover with the top halves of the rolls. Place 2 of the sandwiches on the preheated panini press, close the lid and cook until the cheese is melted, 3 to 4 minutes. Transfer the sandwiches to a cutting board and cut in half. Repeat to cook the remaining sandwiches. Serve immediately.