Mix olive oil, smoked paprika, garlic powder, onion powder, cayenne, salt, and pepper in a bowl. Rub this mixture all over the drumsticks, making sure every piece is well coated. Let the chicken sit for at least 20 minutes so the flavors soak in properly.
Cut the potatoes into even wedges so they cook at the same rate. Toss them in olive oil, paprika, garlic powder, oregano, salt, and pepper until fully coated. Spread them out on a baking sheet in a single layer with space between each piece.
Preheat your oven to 220°C (425°F) and slide the tray in once it's hot. Roast the wedges for 30 to 35 minutes, flipping them halfway through for even color. They should turn golden brown and crisp at the edges when done.
Heat your grill or grill pan to medium-high heat before adding the drumsticks. Cook for about 25 to 30 minutes, turning occasionally so every side gets that char. In the final few minutes, brush on the BBQ sauce so it caramelizes slightly without burning.
Use a meat thermometer to check the thickest part of the drumstick reaches 165°F (74°C). This step matters more than guessing by color, since drumsticks can look done on the outside while still pink near the bone. Once they hit the right temperature, pull them off the heat.
Let the chicken rest for about 5 minutes before serving so the juices redistribute. Arrange the drumsticks and potato wedges together on a large platter. This short resting step makes a noticeable difference in how juicy the chicken turns out.