Pat the drumsticks dry with paper towels. Place each drumstick meat-side up and, using a sharp knife, slice down toward the bone to separate the meat from the bone, keeping the bottom of the meat attached to the bone; open the meat and make small cuts around the bone as needed, then flip each leg and press down to flatten to an even thickness (butterflying).
Season both sides of the butterflied drumsticks evenly with the BBQ rub and rub it into the meat.
Drizzle the olive oil over the chicken and brush or rub to lightly coat to help prevent sticking.
In a small bowl, whisk the BBQ sauce and honey together and set aside.
Preheat a gas or charcoal grill to high, about 500°F (260°C).
Place the butterflied drumsticks meat-side down directly over the heat with the grill lid closed and cook for 5 minutes to develop grill marks.
Flip the drumsticks, brush the meat side with the honey-BBQ mixture, close the lid, and continue grilling for another 5–10 minutes until the internal temperature reaches 165–175°F (74–79°C).
If the exterior is charring before the chicken reaches temperature, move the pieces to indirect heat or a higher grate away from direct flame and continue cooking until done.
Remove the drumsticks from the grill, let rest briefly, and sprinkle with chopped cilantro before serving if desired.