In a mixing bowl, toss the chicken with the salt, cayenne, black pepper, and 1 tablespoon of the rub. Cover and refrigerate for 2 hours.

Heat the shortening to 350ºF. in an electric deep-fryer or deep cast-iron pot. There should be enough shortening for the chicken pieces to immerse.

Whisk together the eggs, buttermilk, and 1 tablespoon of the rub in another shallow bowl.

Combine the flour and the remaining 2 tablespoons rub in another shallow bowl. Roll the chicken in the flour mixture, then in the milk-egg mixture, then again in the flour. Shake off excess batter. Fry 2 to 4 pieces (the number will depend on the size of the frying pot) at a time, 10 to 12 minutes for thighs and drumsticks, 8 to 10 minutes for wings and breasts. When they pop to the surface of the oil and are golden brown, remove and drain on paper towels.

Serve hot or at room temperature.