Add all ingredients (besides the chicken) to a large pot over medium-high heat.
Stir well to combine. Bring mixture to a gentle boil, then add the raw chicken drumsticks.
Turn heat to medium-low, cover, and simmer for about 20-30 minutes or until chicken is just cooked through. Remove chicken to a plate and ladle some of the sauce into a cup.
Prepare your grill by oiling it well and bringing it to HIGH heat.
Once grill is hot, add chicken drumsticks. (The chicken is cooked at this point, you're just wanting to char the skin a bit). Lather some of the extra BBQ Sauce on the chicken as you grill it, and rotate the chicken drumsticks as each side becomes lightly charred. If your grill is hot enough, this should only take a few minutes!