Stir together buttermilk, 3 tsp cayenne, 2 tsp salt, 1 tsp black pepper, ground mustard, and hot sauce. Marinate drumsticks, covered and refrigerated, at least two hours or overnight.
Preheat oven to 250°F and set a baking sheet fitted with a wire rack alongside stove. In a large, heavy bottomed saucepan, add enough oil to come up 2-inches and heat to 360°F using a deep fat thermometer.
Meanwhile, mix flour, remaining 2 tsp cayenne, 2 tsp salt, and ½ tsp black pepper in a shallow bowl. Remove half the chicken from the marinade letting any excess drain off. Thoroughly coat in seasoned flour and tap to shake off loose flour. Using metal tongs, carefully lay chicken in oil. Fry, turning once or twice and maintaining oil temperature, until golden brown and cooked through, about 15 minutes total. Transfer to the prepared baking sheet and place in the oven to keep warm. Repeat with remaining chicken. Sprinkle with salt to serve.