Preheat oven to 400ºF.
Place the chicken breasts in a resealable plastic bag. Close the seal, leaving about an inch unsealed to let air escape.
Pound the chicken with the smooth side of a mallet or a rolling pin until chicken is about ¼” thick. If the chicken breasts are too thick, using a sharp knife, slice the chicken breasts on the bias into smaller thinner pieces before pounding.
Combine flour, salt and pepper on a pie plate or shallow bowl.
Beat the eggs in another pie plate or shallow bowl and set aside.
Grate the cheese and combine with the bread crumbs on another plate.
Coat the chicken breasts on both sides with the flour mixture, shaking off any excess, then cover with the egg, then finish with the bread crumb and cheese mixture. Make sure to press the crumbs into the chicken breasts on both sides.
Place breaded chicken breasts on a baking rack in the refrigerator for 20-30 minutes. This will make for crispy chicken and help keep the breading from falling off when cooking.
Heat the oil in a large heavy skillet or cast iron-skillet over medium heat. Oil should be about ⅛" deep.
Cook chicken breasts, adding more oil if needed, until golden brown, about 2 minutes per side. The chicken will continue to bake off in the oven.
Transfer chicken breasts to a baking sheet and top each breast with some tomato sauce and a few slices of the fresh or grated mozzarella cheese.
Bake in the oven until the chicken is cooked through and the cheese is melted, about 6 to 8 minutes. If using thick chicken breasts, insert an instant-read thermometer into the center of each breast to make sure the chicken is fully cooked. The temperature should read at least 165 degrees F or 74 degrees C.
Remove from the oven and sprinkle with grated Parmigiano Reggiano, Garnish with basil or parsley leaves.