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.
Yield: 4 servings