Shape the lollipops: With a sharp knife, score around each drumstick just below the knuckle, slice through tendons, and push the meat down toward the thick end to form a rounded “lollipop.” Pat dry. Wrap the exposed bone ends with small pieces of foil to keep them clean if you like.
Season: In a large bowl, toss the drumsticks with olive oil, BBQ dry rub, garlic powder, and smoked paprika until evenly coated. Let stand at room temperature for 15–20 minutes while you heat the grill.
Heat the grill: Set up a two-zone fire at 375–400°F (medium-high) with one hot side and one cool side. Clean and oil the grates. If you have a drip pan, set it under the cool zone to catch drips.
Start grilling: Arrange the lollipops on the cool side, meaty ends facing the heat. Cover and cook 25–35 minutes, turning every 10 minutes, until the skin is bronzed and the thickest part reads 165–170°F on an instant-read thermometer.
Make the glaze: While the chicken cooks, stir together BBQ sauce, honey, Worcestershire, apple cider vinegar, Dijon, smoked paprika, and red pepper flakes in a small saucepan. Simmer over medium-low for 3–5 minutes until glossy and slightly thick; keep warm off heat.
Glaze and finish: Brush the lollipops generously with the warm glaze. Move them to the hot side for a brief kiss of direct heat, about 1–2 minutes per side, brushing once more. Bounce them back to the cool zone as needed so the sugars set without scorching. Cook until the centers reach 180–190°F and the glaze looks tacky and shiny, 8–12 minutes total.
Rest and serve: Transfer to a platter and rest 5–8 minutes. Remove the foil from the bones, then serve with any extra warm glaze for dipping.