Place chicken into a gallon-size ziplock bag. Combine all ingredients for the brine and pour into bag with chicken. Seal tightly and refrigerate for up to 2 hours (at least a minimum of 30 minutes).
Remove chicken from bag and pat dry with paper towels, discarding the brine.
Oil grill grates and heat to medium high. Place chicken on grill and cook about 10 minutes on each side to crisp up the skin and get nice grill marks.
Bake chicken in a 375 degree oven to finish cooking, about 35-40 minutes or until chicken reaches an internal temperature of 170 degrees. After 20 minutes, baste all over with barbecue sauce. Baste again before the end of cooking.
Serve with extra sauce.
In a medium saucepan, stir together all the ingredients except the oil. Bring to a boil, then reduce the heat and simmer for 25 minutes. Remove from the heat and whisk in the oil until smooth.
Poultry is considered safe to eat when the internal temperature reaches 165 degrees F. Because the bones can affect your temperature reading, bake until a thermometer reads 170 degrees F to be safe.
Let oven barbecue chicken rest for 10-15 minutes before serving. This allows the juices to distribute evenly through the meat instead of leaking out onto your plate!