In a large bowl, dissolve 1/4 cup of kosher salt in 2 quarts of cold water. Add the chicken and refrigerate for 2 hours.
Meanwhile, melt the butter in a medium saucepan. Add the onion and garlic and cook over low heat until softened. Add the paprika, ancho powder and crushed red pepper; cook, stirring, for 1 minute. Add 1 cup of water, the brown sugar and vinegar and bring to a simmer. Stir in the tomato paste and molasses and simmer over low heat, stirring occasionally, until the sauce is thickened, about 10 minutes. Season with salt and let cool.
Light a grill or preheat the oven to 500°. Drain the chicken and pat dry, then grill over a moderately low fire, turning a few times, until just cooked through; about 25 minutes for the white meat; about 35 minutes for the dark. Alternatively, in the upper third of the oven, roast the chicken, skin side up, on a rimmed baking sheet, until just cooked through; 25 minutes for the white meat; 35 to 40 minutes for the dark.
Set aside half of the barbecue sauce; brush the remaining sauce on the chicken. Continue grilling or roasting the chicken for 10 minutes, or until nicely glazed. Serve with the reserved sauce.
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