Pound chicken until 1/3-inch thick. Season with salt and pepper.
Dip chicken in the beaten eggs, then roll in breadcrumbs.
Heat oil in a large oven-safe skillet. When oil is hot, brown chicken on all sides. Remove and set aside.
Preheat oven to 350°F.
Discard excess oil from pan, leaving about 1 tablespoon of oil in the skillet. Sauté chopped onion and garlic until onions are slightly translucent. Stir in tomato sauce, drop in a sprig of basil, and bring to simmer for 10 minutes. Remove the basil.
Place chicken in the sauce. Sprinkle Parmesan over the chicken and place a round of mozzarella on top of each chicken breast. Transfer to oven and bake at 350°F for 15 minutes, or until cheese has melted and browned.