Preheat the oven to 400 degrees F (200 degrees C). Line a baking sheet with foil.

Mix olive oil, salt, pepper, paprika, garlic powder, onion powder and cayenne together in a large bowl. Add drumsticks and toss to coat. Spread out on the prepared baking sheet.

Bake in the preheated oven for 20 minutes. Remove from oven and brush with barbeque sauce; bake for 7 to 8 minutes. Flip drumsticks, brush again with barbeque sauce, and bake an additional 7 to 8 minutes.

Set an oven rack about 6 inches from the heat source and turn on the oven's broiler.

Brush drumsticks with more barbeque sauce and broil for 3 to 4 minutes. Flip drumsticks, brush with barbeque sauce, and broil until chicken is no longer pink at the bone and the juices run clear, 3 to 4 more minutes. An instant-read thermometer inserted near the bone should read 165 degrees F (74 degrees C).