Pat the drumsticks dry with paper towels.
Using a sharp knife on a cutting board, cut around the bone where the thicker meat begins and trim away skin and meat to expose 2–3 inches of bone to create the "lollipop" shape; trim a small piece from the bottom if needed so each will stand upright.
Prepare the marinade by whisking together chopped basil, 1 cup of the olive oil, garlic salt, cracked black pepper, smoked paprika, and red pepper flakes in a mixing bowl.
Coat the prepared chicken lollipops thoroughly with the marinade (or rub with BBQ seasoning if using) and refrigerate to marinate for at least 30 minutes or up to overnight.
Preheat your pellet smoker (such as a Traeger) to 250°F (121°C).
Arrange the marinated lollipops on the smoker rack and smoke at 250°F for about 1.5 hours, or until the internal temperature reaches approximately 160°F.
Increase the smoker temperature to 375°F (190°C). Brush or spritz the lollipops with the remaining olive oil to help crisp the skin.
Continue cooking at 375°F until the internal temperature reaches 165°F and the skin is crisp and browned to your liking, checking with a meat thermometer.
Remove the chicken lollipops from the smoker and let rest for a few minutes before serving.