Space oven racks equal distance in oven. You will be using both racks for this recipe. Preheat oven to 425° F. Grab two baking sheets and line them with parchment paper. Set aside.
Place prepared sliced carrots and onions on one of the baking sheets and drizzle with 1 tablespoon olive oil. Season with salt and black pepper, to taste, and toss to coat. Spread vegetables into a single layer and set aside.
Place a metal baking rack (cooling rack) on the other baking sheet and spray with non-stick cooking spray.
Pat drumsticks completely dry with paper towels. Sprinkle on all sides with 2 teaspoons smoked paprika, 2 teaspoons garlic salt, 2 teaspoons onion salt, and 2 teaspoons black pepper and arrange seasoned drumsticks on baking rack.
Place both baking sheets into preheated 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 at least 165°F and the vegetables are tender and caramelized. (Depending on your oven and how thin you sliced the vegetables, they may be done a little sooner).
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 roasted root vegetables.