Bring the water to a boil. Stir in the remaining brine ingredients. Make sure the brine is cold before adding the chicken.
Place the chicken, breast side down, on a work surface. Cut from the tail to the neck along both sides of the backbone using poultry shears. Remove the backbone.
Use the shears to cut through the cartilage at the tip of the breastbone about a quarter of an inch. Bend the breastbone open by pressing firmly on the center of the breastbone.
Season the chicken all over with the rub.
Place the chicken in a large, resealable plastic bag and pour in the brine. Press the air out of the bag and seal tightly. Turn the bag to distribute the brine, place in a bowl, and refrigerate for 2 hours.
Set the grill to (400°F). Preheat for at 15 minutes and brush the grill grates clean.
Remove the chicken from the bag and discard the liquid. Pat the chicken dry with paper towels and lightly brush the it with oil.
In a saucepan combine all the glaze ingredients. Cook over medium heat on the stove until reduced and sticky, 10 to 15 minutes, stirring occasionally.
Place the chicken, skin side up, on the grill, and close the lid. Brush the top of the chicken every 15 minutes with the glaze.
Once the internal temperature reaches 160°F, remove the chicken from the grill with tongs.
Rest the chicken at room temperature, indoors, for 10 minutes before carving. The temperature of the chicken will continue to rise as it rests.
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