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To make the South Carolina-style barbecue sauce, heat olive oil in a medium-sized saucepan over medium heat. Add the minced onions and sauté for 10 minutes. Edge heat down slightly if onions start to brown. Add the minced garlic and sauté for 1 minute more.
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Next, stir in all of the remaining ingredients. Bring mixture to a gentle boil and then turn heat to low. Stirring often, cook over low heat for 10 minutes. Adjust sweeteners, spices, vinegar, and salt levels to taste.
Spoon half the sauce into a bowl for brushing on the grilled chicken, and reserve remaining sauce for serving.
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Sprinkle both sides of the chicken with sea salt.
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Get a large pile of charcoal going in your grill. When all the coals are gray, use tongs to move them to one side of the grill.
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Brush chicken on both sides with barbecue sauce and place it on the grill over the coals. (Keep an eye out for flare ups –– move chicken away from the coals for a little while if they get really intense). After 2 – 3 minutes, flip chicken.
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After another 2 – 3 minutes move chicken to the side of the grill that’s not over the coals and brush with barbecue sauce again. Cover grill and wait 4 minutes or so. Flip the chicken, brush with sauce again, and cover grill for another 4 minutes or so.
Pull chicken from grill when the outside is a deep golden brown and the internal temperature reaches 165 degrees F.
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Rest chicken for 10 minutes before serving.
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Serve chicken with a generous drizzle of barbecue sauce.