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Dissolve 1/2 cup hot sauce, 1/4 cup salt, and sugar in 2 quarts cold water in a large container. Add chicken and refrigerate, covered, for at least 1 hour or up to 3 hours.
Set a wire rack in a rimmed baking sheet. Whisk flour, granulated garlic, paprika, cayenne, one tablespoon pepper, and one teaspoon salt together in a large bowl. Remove chicken from brine. Working in batches of four, dredge chicken pieces in flour mixture, pressing to adhere. Place chicken on prepared rack. Refrigerate chicken, uncovered, for at least 30 minutes or up to 2 hours.
FOR A CHARCOAL GRILL: Open bottom vent completely. Light a large chimney starter mounded with charcoal briquettes (7 quarts). When top coals are partially covered with ash, pour into a steeply banked pile against the side of the grill. Set cooking grate in place, cover, and open lid vent completely. Heat grill until hot, about 5 minutes.
FOR A GAS GRILL: Turn all burners to high, cover, and heat grill until hot, about 15 minutes. Turn the primary burner to high and turn off other burner(s). (Adjust primary burner [or, if using three-burner grill, primary burner and second burner] as needed to maintain grill temperature of 425 degrees.)
Clean and oil the cooking grate. Place chicken, fatty side up, on the cooler side of the grill, arranging drumettes closest to the coals. Cook chicken, covered, until lightly browned and coating is set, about 30 minutes for charcoal or about 45 minutes for gas.
Brush chicken with oil until no traces of flour remain (use all oil). Cover and continue to cook until coating is golden brown and chicken registers between 180 and 200 degrees, about 30 minutes longer for charcoal or about 45 minutes longer for gas. Transfer chicken to a clean wire rack and let cool for 10 minutes.
Combine butter and remaining 1/2 cup hot sauce in a large bowl and microwave, covered, until butter is melted, about 1 minute. Whisk to combine fully. Add chicken and toss to coat. Serve.