Combine all ingredients except the chicken and whisk together in a small saucepan. Turn heat on medium high and bring to a boil. Boil for about 5 minutes, then turn heat down and simmer another 5 minutes. Set aside 1/4 cup of the marinade and store in the fridge.
Let cool slightly, then combine the rest of the marinade and chicken drumsticks in an airtight container. Allow chicken to marinade at least 4 hours, turning occasionally.
Preheat grill. Place drumsticks on grill set to medium heat. Drumsticks require a longer cooking time to cook thoroughly so make sure your grill is set to medium! Grill for about 35-40 minutes, turning once every 7-8 minutes. Lightly brush drumsticks once or twice while cooking, using the 1/4 cup of the marinade that was set aside previously. Drumsticks should reach an internal temperature of 185 degrees F. Remove chicken from the grill and allow them to rest 10 minutes prior to serving.