Meaty chicken leg quarters stay moister on the grill than chicken breasts and are a perfect vehicle for the sauce. The thick sauce glazes the meat, making for delightfully sticky-fingered eating. Substitute any other wood chips for hickory.
Loosen skin from thighs and drumsticks by inserting fingers, gently pushing between skin and meat. Rub Kansas City Dry Rub evenly under loosened skin; let stand at room temperature for 30 minutes.
Remove grill rack; set aside. Prepare grill for indirect grilling, heating one side to medium-high and leaving one side with no heat.
Pierce bottom of a disposable aluminum foil pan several times with the tip of a knife. Place pan on heated side of grill; add half of wood chips to pan. Place another disposable aluminum foil pan (do not pierce pan) on unheated side of grill. Pour 2 cups water in pan. Coat grill rack with cooking spray; place grill rack on grill.
Place chicken on grill rack over foil pan on unheated side. Close lid; cook 1 1/2 hours. Add additional wood chips halfway during cooking time. Turn chicken over; cover and cook 30 minutes or until a thermometer registers 165°. Remove chicken from grill; let stand 10 minutes. Remove and discard skin.
Bring sauce to a simmer in a small saucepan. Brush chicken with 1/2 cup sauce. Serve chicken with remaining sauce.
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