In a small bowl, combine the chili powder, salt, pepper, garlic powder, smoked paprika, onion powder, and cayenne pepper.
Pat the chicken drumsticks dry with paper towels.
Drizzle with olive oil and coat evenly with the spice blend.
For a gas grill, preheat to 400°F (204°C). Place the chicken on clean, lightly oiled grates.
Grill over direct heat, turning every few minutes until the skin becomes golden brown and crispy.
Reduce the grill temperature to about 300°F (149°C), close the lid, and continue cooking for about 25 minutes, turning halfway through.
For a charcoal grill, create two cooking zones. Brown the chicken over direct heat before moving it to indirect heat to finish cooking.
Brush each piece generously with BBQ sauce during the final few minutes of cooking.
Cook until the BBQ Chicken Drumsticks reach an internal temperature of 165°F (74°C).
Rest for five minutes before serving.