Heat grill to medium. Simmer sauce ingredients (first 7 ingredients listed above) in a small saucepan over medium heat until reduced to about 1 1/4 cups, 5 to 7 minutes. Divide barbecue sauce between two bowls; use one for basting and set aside the other for serving.
In a large bowl, toss chicken with vegetable oil; season well with coarse salt and ground pepper.
Lightly oil grates; place chicken on grill. Cover grill; cook, turning frequently, until chicken registers 165 degrees (160 degrees for breast) on an instant-read thermometer, 10 to 15 minutes. Uncover grill; continue cooking, basting frequently with sauce, until chicken is glazed thoroughly, 3 to 4 minutes more. Serve with remaining sauce.