Preheat your oven to 400°F (200°C) if you’re baking. If you’re grilling, heat the grill to medium-high and oil the grates lightly.
Pat the drumsticks dry with paper towels. This helps the glaze cling and the skin crisp. Toss them with olive oil, salt, pepper, paprika, and garlic powder until evenly seasoned.
In a small bowl, whisk together BBQ sauce, honey, and soy sauce. This will be your glaze. Taste it if you like a touch more sweetness or tang.
Arrange the drumsticks on a lined baking sheet or on a lightly oiled grill rack. Leave a bit of space between each piece so they cook evenly.
If baking: bake for 25 minutes, then brush on half of the glaze. Return to the oven for another 5–10 minutes, until the internal temperature reaches 165°F (74°C) and the skin is caramelized.
If grilling: Grill for about 10–12 minutes, then brush with glaze. Flip and grill another 8–10 minutes, brushing with glaze every few minutes until the glaze is glossy and sticky and the chicken is cooked through.
Let the drumsticks rest for 5 minutes before serving. A final brush of glaze right before serving adds a nice shine.