Combine everything but the chicken in a large glass bowl. Reserve ½ cup of the marinade to use as a baste.
Add the chicken to the marinade a few pieces at a time, turning until coated.
Transfer to 1-gallon resealable plastic bags or covered plastic containers. Divide any remaining marinade evenly among the bags or containers of chicken, and place in the refrigerator to marinate for at least 4 hours. (This can even be done 2 days in advance.)
Preheat the oven to 350°F.
Lightly oil a rimmed baking sheet. Arrange the chicken skin side up on the baking sheet and roast for 35 minutes. The chicken will be finished on the grill; it can be roasted up to a few hours in advance, or grilled right after it comes out of the oven.
Preheat the grill to high (or glowing embers if using charcoal).
Place the chicken skin side up on the grill grate. Using a long-handled barbecue brush, brush the top of the chicken with the reserved baste.
Grill for 2 minutes, turn, and grill skin side down for 2 minutes.
Carefully remove the chicken from the grill with tongs (the chicken skin may stick to the grate) and serve hot.