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1
Place the brown sugar, soy sauce, sake or sherry, and garlic in a large resealable plastic bag and stir to combine.
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2
Add the chicken, seal the bag (pressing out any excess air), and turn to coat the chicken evenly.
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3
Place in the refrigerator and let marinate, turning occasionally, for at least 2 hours or up to 24 hours.
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4
Place all of the ingredients except the cornstarch-water mixture in a small saucepan and stir to combine.
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5
Bring to a boil over medium-high heat.
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6
Reduce the heat to low and simmer until the sauce has reduced to about 1 1/4 cups, about 15 minutes.
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7
Add the cornstarch mixture, whisk to combine, and cook until the sauce has thickened, about 1 minute.
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8
Divide the sauce equally between 2 small bowls; set aside 1 bowl for serving and the other for brushing onto the chicken while it cooks on the grill.
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9
When ready to grill, remove the chicken from the refrigerator and let it sit at room temperature for about 15 minutes.
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10
Meanwhile, heat a gas or charcoal grill to medium (about 350 degrees F to 450 degrees F).
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11
When the grill is ready, rub the grates with a towel dipped in vegetable oil.
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12
Remove the chicken from the bag and place it on the grill (skin-side down if using skin-on), cover the grill, and cook until grill marks appear on the bottom, about 5 minutes.
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13
Flip the chicken pieces, cover the grill, and cook until grill marks appear on the second side, about 5 minutes more.
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14
Flip the chicken again and generously brush the pieces using the bowl of sauce set aside for that purpose.
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15
Cover the grill and cook for 5 minutes.
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16
Flip and brush the chicken, and continue to flip and brush every 5 minutes, until an instant-read thermometer inserted into the thickest piece reads 160 degrees F to 165 degrees F and the meat near the bone is no longer pink, about 10 to 15 minutes more (about 25 to 30 minutes total cooking time).
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17
Remove to a clean serving platter, sprinkle with the toasted sesame seeds, and serve, passing the remaining bowl of barbecue sauce on the side.