1. Place all sauce ingredients into blender and blend until smooth. Place drumsticks in a large freezer bag and pour three-quarters of sauce over them. Seal bag and place in refrigerator for two hours or more. Reserve remaining sauce in a container for later.
2. When ready to cook, preheat grill to 500 degrees Fahrenheit. Lower flame to medium. Turn off one burner and place chicken skin-side down on the grill over the burner that is turned off. Close the lid of the grill and cook chicken for 10 to 15 minutes. Flip over and repeat on the other side for an additional 10 to 15 minutes, brushing with reserved sauce two to three minutes before done.
Bake chicken in oven, covered, at 400 degrees Fahrenheit for 45 minutes. Then, either uncover and bake another 20 minutes until crispy, brushing with reserved sauce a few minutes before done, or transfer to grill, brush with reserved sauce, and grill on high for two to three minutes per side.