Preheat oven to 425°. Place chicken breast halves in a zip-top plastic bag. Combine buttermilk and egg. Add egg mixture to bag, and seal. Marinate in refrigerator for 4 hours. Remove chicken from bag; discard marinade. Sprinkle chicken with salt and pepper. Combine flour and ground oats. Dredge the chicken in flour mixture. Heat oil in a large ovenproof skillet over medium-high heat. Add chicken; sauté for 4 minutes. Turn chicken over. Bake at 425° for 10 minutes or until done.