Line one large or two rimmed baking sheets with parchment paper, baking mats, or aluminum foil and set aside.
Place carrots on one of the prepared baking sheets (or one half if using one pan) and drizzle with olive oil. Season with salt and pepper, to taste, and toss to coat. Spread into a single layer and set aside.
In a small bowl combine smoked paprika, garlic powder, onion powder, salt and pepper.
Pat drumsticks completely dry with paper towels. Sprinkle on all sides with seasoning mix and arrange seasoned drumsticks on second baking sheet (or second half).
Place both baking sheets into pre-heated oven, with chicken on the upper rack.
Roast for 20 minutes, then rotate both baking sheets and roast for another 15-20 minutes, or until an instant-read thermometer inserted into the chicken reads 165°F and the vegetables are tender and caramelized. (Time will vary depending on your oven and how thick your carrots and drumsticks are).
Remove each baking sheet from oven when ready and transfer chicken to a serving platter. Loosely cover chicken with aluminum foil and let rest for 5 minutes before serving with carrots.