Heat the oven to 425°F and line an 18-by-13-inch rimmed sheet pan with foil. Pat drumsticks dry and coat with 1 tablespoon oil, paprika, garlic powder, pepper and 1/2 teaspoon salt.
Toss sweet potato wedges with remaining oil and salt. Arrange chicken and potatoes in separate areas of the pan in one layer.
Roast for 25 minutes, then turn the drumsticks and potato wedges. Roast for 10 minutes more and transfer any fully tender potatoes to a clean plate.
Brush drumsticks with barbecue sauce and roast for 8 to 12 minutes until the sauce is tacky and chicken reaches at least 165°F, preferably 175 to 185°F for tender dark meat.
Rest for 5 minutes, returning any reserved potatoes to a clean open section of the pan for the last 2 minutes of cooking if they need warming. Serve 2 drumsticks with one-quarter of the potatoes per portion.