Remove chicken from the marinade and allow to drain on a wire rack set over a rimmed baking sheet for 1 hour before cooking (discard marinade). Meanwhile, whisk together the flour, cornmeal, and remaining 1/4 teaspoon cayenne; season with salt and pepper. Spread mixture in a shallow dish.
When you are ready to begin frying, pour a scant 1/2 inch oil into a large cast-Iron skillet and heat over medium until oil registers 375 degrees F on a deep-fry thermometer. (Alternatively, test by dropping a cube of white crustless bread into the oil; it should turn golden brown within 1 minute.)
While the oil is heating, and working with a few parts at a time, dredge chicken in the flour mixture, turning to completely coat. Shake off excess flour and set chicken on a parchment-lined baking sheet as you work.
Preheat oven to 200 degrees F. Set a clean wire rack on a rimmed baking sheet with several layers of paper towels on top of rack. Working in batches (skillet should be filled but without pieces touching each other), arrange chicken, skin side down, in a single layer. Adjust heat so temperature of oil remains between 330 degrees F and 340 degrees F during frying. Cover and cook until chicken is crisp and golden on bottom and parts remove easily from pan, 4 to 5 minutes. Carefully turn chicken and continue frying (covered) until crisp and cooked through (breasts should register 160 degrees F and thighs 165 degrees F on an instant-read thermometer), 4 to 5 minutes more. Remove each part as soon as it is ready (wings, drumsticks, and thinner breast pieces cook faster than thighs). Transfer to prepared rack on baking sheet and keep warm in the oven while cooking remaining chicken, returning oil to 375 degrees F before adding each batch. Serve chicken hot.