In a resealable plastic bag, soak the chicken in the buttermilk for 30 minutes (or up to 12 hours). In a medium bowl, combine the flour, salt and pepper.
Remove the chicken pieces and coat them evenly with the flour mixture; transfer to a parchment-lined baking sheet and let stand for 15 minutes. Coat the chicken again in the flour mixture.
Meanwhile, in an extra-large, heavy skillet, heat the oil to 350°. Carefully add the chicken to the pan and partially cover. Lower the heat to medium-low and fry the chicken, turning occasionally, until the crust is gold-brown and the meat is cooked through, 25 to 30 minutes. (Lower the temperature if the chicken is browning too quickly.) Transfer to paper towels to drain; serve hot.