Preheat the oven to 425°F. Line a large sheet pan with foil or parchment and lightly grease it.
In a small bowl, mix the smoked paprika, garlic powder, onion powder, salt, and pepper.
Toss the potatoes with 1 tablespoon of oil and half the spice mix. Spread them on one side of the pan.
Pat the drumsticks dry, rub with the remaining oil and spice mix, and arrange on the other side of the pan.
Roast for 25 minutes. Flip the potatoes and brush the drumsticks generously with barbecue sauce.
Roast 15–20 minutes more, brushing with more sauce halfway, until the chicken reaches 175°F inside and the potatoes are golden and tender.
Rest 5 minutes and serve with extra barbecue sauce.