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Mix brown sugar, paprika, black pepper, cumin and cayenne pepper in medium bowl. Place ribs on large baking sheet and rub ribs with salt. Sprinkle spice mixture evenly all over ribs and massage spices into meat. Cover and refrigerate overnight.
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Prepare charcoal or gas barbecue for low heat (250°F). For 2-burner gas grill, turn 1 burner off and for 3-burner gas grill, turn left and center burners off, leaving right burner on. Place aluminum pan over unlit burner. For a charcoal grill, pile charcoal on one side of barbecue and place aluminum pan on opposite side of barbecue. Add about 1 cup of water to aluminum pan.
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Drain water from wood chips and sprinkle them over charcoal or place them in wood chip holder of barbecue or in another broiler pan set it alongside pan of water. Combine vinegar and 3/4 cup of water in a clean spray bottle.
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Place ribs on grill rack over unlit burners or opposite of charcoal. Placing ribs on opposite side of heat source ensures they don’t char too much.
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Close hood of barbecue and grill ribs until meat around bones is very tender, turning ribs over as needed and spraying ribs occasionally with cider-water mixture, about 3 hours. It may be necessary to occasionally open barbecue lid or slightly prop lid open to regulate temperature from getting too high.
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Once ribs are tender, begin brushing them lightly with barbecue sauce, allowing sauce to set a bit before applying next coat. Continue brushing ribs with barbecue sauce, turning as needed, until meat begins to pull away from bone easily, about 30 minutes.
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Transfer ribs to carving board and let rest for about 5 minutes. Using large sharp knife, cut racks into individual ribs and toss ribs in large bowl with enough of remaining warm barbecue sauce to coat.
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Arrange ribs on a platter and serve remaining warm barbecue sauce alongside.