Put the chicken in a zip top bag with the buttermilk marinade ingredients: the buttermilk, Dijon mustard, 2 teaspoons of salt, 1 teaspoon of black pepper, and cayenne. Massage everything together, seal the bag, and refrigerate for 8 hours or overnight.
When ready to fry, set two wire racks over two baking sheets. Mix together the flour, cornmeal, salt, and pepper in a paper bag or a large bowl. Remove the chicken pieces from the buttermilk brine and coat them with the flour mixture, working a few pieces at a time, making sure the pieces are thoroughly coated.
In a large, heavy skillet, such as cast iron, heat about 1 1/4 inch of duck fat or canola oil. The temperature should read 350ºF (180ºC) when ready to fry.
Carefully place as many pieces of the chicken as will fit in the pan without crowding it, skin side down. The temperature will decrease but try to keep the temperature of the oil around 325ºF (162ºC) as you fry.
Fry the chicken until the bottom is nicely browned, then use tongs to turn the chicken over and fry until the other side is browned and crisp. The chicken will take about 20 to 25 minutes to cook.
As the chicken is done, remove the pieces and set them on the wire rack. Use a slotted spoon to skin the fat or oil of stray pieces of chicken, so they don’t burn, then fry the remaining pieces of chicken.
