Rinse the chicken and pat it dry. Place the chicken in a large glass bowl and pour the buttermilk over it. Cover and refrigerate for 30 minutes. In a medium bowl, mix the flour, cornmeal, salts, peppers, and paprika. Dredge the chicken pieces in the flour mixture, turning to coat all sides thoroughly. Rest the dredged chicken for 20 to 30 minutes, to allow the buttermilk to absorb the batter. In a large cast-iron skillet, add the oil to a depth of three-quarters of an inch. Over medium-high heat, warm the oil to 350°F, using a kitchen thermometer to test the temperature. Carefully put the chicken (a couple of pieces at a time), skin side down, into the oil. Reduce the heat to medium and cook for 15 minutes, or until nicely browned. Turn the chicken and cook for an additional 10 minutes, or until the internal temperature registers 180°F on a meat thermometer. Remove the chicken and drain it on paper towels.