Mix all of the spice rub ingredients together in a small bowl. Put the drumsticks in a large bowl and rub the spice all over the chicken until the chicken is fully coated. Cover with plastic wrap and let it sit in the fridge for 30 minutes. (Be sure to thoroughly wash your hands after handling raw chicken!!!!)
Fire up the grill and cook the chicken until a thermometer registers an internal temperature of 165 °F in the thickest part of the chicken drumstick. Towards the end of cooking, brush on a thin layer of BBQ sauce.