Combine the buttermilk, Dijon mustard, 2 teaspoons salt, 1 teaspoon black pepper, and cayenne in a large zip-top bag or bowl.
Add the chicken pieces to the buttermilk mixture, massage to coat, seal the bag, and refrigerate for 8 hours or overnight.
When ready to fry, set two wire racks over two baking sheets to drain cooked chicken.
In a large paper bag or bowl, whisk together the flour, cornmeal, 2 teaspoons salt, and freshly ground black pepper.
Working in batches, remove chicken from the buttermilk, let excess drip off, then toss or shake pieces in the flour mixture to thoroughly coat. Place coated pieces on a wire rack and gently shake off excess flour.
Heat about 1 1/4-inch (3–4 cm) of duck fat or canola oil in a large heavy skillet over medium heat. Heat until the oil reaches about 350°F (180°C) on a thermometer.
Carefully add as many chicken pieces as will fit without crowding, skin side down. The oil temperature will drop; aim to maintain about 325°F (162°C) while frying.
Fry until the bottom is nicely browned, then turn with tongs and fry until the other side is browned and the chicken is cooked through, about 20–25 minutes total per batch. Check the thigh near the bone: juices should run clear.
Remove cooked pieces to the wire rack. Use a slotted spoon or skimmer to remove stray bits from the oil between batches to prevent burning, then continue frying remaining chicken.