Preheat the 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 baking sheet.
Make the seasoning paste: In a medium bowl, use a fork or spatula to combine 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 for about 40 minutes, until their internal temperature, measured with an instant-read thermometer not touching a bone, reaches 165°F.
Baste the drumsticks with the pan juices using a clean pastry brush. Serve immediately.
Video
Notes
Check the drumsticks after the first 30 minutes. If they seem to be browning too much on top, loosely cover them with foil and keep cooking until they are cooked through.
When I'm lazy, instead of making a seasoning mix, I arrange the drumsticks in the pan, spray them with olive oil, and sprinkle them with the spices.
If your packages have five drumsticks each, you can cook ten drumsticks using the same recipe, increasing the olive oil by half a tablespoon.
The cayenne greatly improves the flavor of this dish. It adds a hint of spice without overwhelming the chicken. But if you're not a fan of spicy foods, use just ⅛ teaspoon (a pinch). It's also OK to omit it completely.
You can keep the leftovers in an airtight container in the fridge for 3-4 days. Reheat them uncovered in a 350°F oven to crisp the skin, or enjoy them cold! I sometimes remove the skin, cut the chicken into chunks, and use it in a salad.