Preheat your oven to 400°F. Line a rimmed baking sheet with high-heat-resistant parchment paper and arrange the drumsticks in a single layer on the parchment paper.
Make the seasoning paste: In a medium bowl, use a fork or a spatula to mix the olive oil, kosher salt, black pepper, garlic powder, onion powder, smoked paprika, and cayenne pepper.
Using a pastry brush or your hands, coat the chicken pieces with the seasoning paste.
Bake the drumsticks uncovered until their internal temperature, measured with an instant-read thermometer and not touching a bone, reaches 165°F, about 40 minutes.
Baste the drumsticks with the pan juices using a clean pastry brush. Serve immediately.
