Pat the chicken dry, then use your hands to toss with remaining 1 tablespoon oil, salt, and pepper to coat.
Transfer drumsticks to the hot grill and cook, covered, for 20-30 minutes, turning after about 15 minutes or until the skin is golden brown. Once the drumsticks are cooked through (170-190F), you can reduce the heat a bit to prevent burning before adding the BBQ sauce.
Brush them with the BBQ sauce on one side, cover and grill for 3 minutes. You want a little bit of char, but not too much, so adjust your grill accordingly. Flip and brush the other side with more sauce, then cover and grill for 2-3 more minutes.
Transfer them a serving platter to let them cool off for a few minutes. If you see any naked spots, brush some more BBQ sauce on them. Serve warm!