The day before, make a dry brine: Put 2 wire racks on 2 rimmed baking sheets. Mix together the sugar, chili powder, mustard powder, garlic powder, cayenne and 1/2 cup salt. Pat the turkey drumsticks dry. Put 2 drumsticks in a large shallow bowl and sprinkle generously with the dry brine, rubbing it into every nook and cranny. Transfer to one of the wire racks. Working in batches, sprinkle and rub the remaining drumsticks with the dry brine and transfer to the wire racks. Sprinkle the drumsticks on the racks with any remaining dry brine. Refrigerate, uncovered, for at least 8 hours and preferably up to 24 hours.
Preheat the oven to 350 degrees F. Bake the drumsticks, rotating the baking sheets front to back and top to bottom halfway through cooking, until golden brown and an instant-read thermometer measures at least 200 degrees F, about 2 hours. Let cool to room temperature. Transfer the drumsticks to large re-sealable plastic bags. (The turkey can be made ahead up to this point and refrigerated for up to 24 hours.)
Preheat a gas grill to medium-low or prepare a charcoal for medium-low indirect heat: bank one chimney starter-full of lit and ashed-over charcoal briquettes to one side of the grill. Set up a drip pan on the other side to avoid flare-ups. (Be sure to consult the grill manufacturer's guide for best results).
Brush the grill grates with oil. Put the drumsticks on the grill (or indirect side if using charcoal) and cook, turning and basting with the barbecue sauce every 5 minutes, until the drumsticks are deep amber on the outside and heated through, about 15 minutes.
Let cool for 5 minutes, then wrap the exposed bone-end of each drumstick with foil and serve while warm.
If your turkey legs are on the smaller side, dry brine for 6 to 8 hours, not overnight.
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