Season the chicken breasts on both sides with salt and pepper. Place the flour, egg and panko crumbs into separate shallow dishes. Coat the chicken breasts in flour, shaking off any excess. Dip them into the egg, and then press into the panko crumbs until well coated on both sides.
Heat 1/4 inch of oil in a large skillet over medium-high heat. Place chicken in the hot oil, and cook 3 or 4 minutes per side, or until golden brown.