Brine chicken. Stir together water, salt, and sugar in a large bowl and add chicken or transfer brine to an extra-large 2.5-gallon Ziploc bag, add chicken, and seal. Place chicken and brine in the fridge for one hour.
Make barbecue sauce. While the chicken brines, whisk together the barbecue sauce ingredients in a medium saucepan and heat slowly on low heat for about 15 minutes.
Grill chicken. Drain water and pat the chicken dry. Preheat a gas grill to medium heat and the oven to 375℉. Pour a little bit of oil on folded paper towels and using large tongs, use the folded paper towel to quickly grease the grates of the grill. Repeat if necessary. Place chicken on the grill and cook for 10 minutes, turning halfway through.
Bake chicken and add sauce. Transfer chicken to a lined baking sheet. Bake chicken for 15 minutes, remove from the oven, and brush barbecue sauce all over the chicken. Return chicken to the oven and cook for 15 minutes, then remove chicken from the oven again and brush barbecue sauce on both sides of the chicken. Finish cooking the chicken for an additional 15 minutes or until the chicken is cooked to 165℉, as checked with a meat thermometer.
Serve. Brush chicken with additional sauce and serve immediately.