Place chicken between sheets of waxed paper and pound with a rolling pin or flat side of a meat mallet to 1/4-inch thickness (see Tip Strip on page 104).
Toast bread crumbs in a dry, ovenproof skillet over medium heat until golden, shaking pan occasionally to stir, 2 to 3 minutes. Transfer bread crumbs to a bowl and let cool. Stir in Parmesan and basil and set aside.
Sprinkle chicken with salt and pepper and dredge both sides in flour. Warm oil in skillet over medium-high heat and cook chicken until brown and cooked through, turning once, about 3 minutes per side.
Preheat broiler to high. Remove skillet from heat and spoon tomato sauce over chicken. Sprinkle with bread crumb mixture and broil 1 minute until just toasted. Sprinkle with mozzarella; broil until cheese is golden and bubbling, 1 to 2 minutes longer, and serve.