Place chicken pieces into a plastic container and cover with buttermilk. Cover and refrigerate for 12 to 24 hours, turning the meat at least once.
Add enough shortening to come 1/4 inch up the sides of a 12-inch cast-iron skillet. Set over low heat. Once shortening liquefies, raise heat to reach 325ºF on candy/fry thermometer. Do not allow oil to exceed 325ºF.
Drain chicken in a colander. Combine salt, paprika, garlic powder, and cayenne pepper. Liberally season chicken with spice mixture.
Put the flour in a bag or container with a lid and dredge the chicken in the flour. Shake off the excess and set aside for 2 minutes before frying.
Place chicken skin-side down in the pan. Put thighs in the center, and breast and legs around the edge of the pan. The oil should come halfway up the pan. Cook chicken until golden brown on each side, 10 to 12 minutes per side. More importantly, the internal temperature should be right around 180ºF. (Be careful to monitor shortening temperature every few minutes.)
Drain chicken on a wire rack set over a paper towel-lined half-sheet pan. Let rest 15 to 20 minutes before serving. If you need to hold the chicken before serving, cover loosely with foil but avoid holding in a warm oven.