Line a rimmed sheet pan with foil or parchment paper; set pan aside.
Pat the chicken drumsticks dry with a paper towel and place drumsticks in a large bowl.
Using a small bowl, combine the smoked paprika, salt, black pepper, garlic powder and onion powder. Add spice mixture to the bowl with the drumsticks; toss until well coated.
Using a small bowl, whisk together the olive oil, thyme, rosemary and garlic. Drizzle this mixture evenly over the drumsticks and toss until well coated.
Bake for 20 minutes. Flip drumsticks over and continue baking for 20-25 more minutes, or until internal temperature reaches 165°F. (Note: Make sure the meat thermometer is not touching bone as that will give an inaccurate reading.)
Turn broiler on high and broil for 2-3 minutes per side, or until drumsticks are golden brown and crispy.