Prepare the brine solution of water, salt and sugar. Double or half the ingredients for smaller or larger batches depending on how many turkey legs you are preparing. You will need just enough to cover the turkey drumsticks completely. Make sure to mix the salt and sugar in completely so that they are totally dissolved before adding the turkey leg.
In a large container pour the brine solution over your turkey drumsticks so that they are completely covered. If necessary you can use a plate or bowl to weigh the legs down to keep them submerged.
Refrigerate for at least 4 hours and up to 24 hours, preferably overnight. Take out about 1 hour before cooking to bring to room temperature.
Remove the turkey leg from the brine and pat dry.
Just before cooking, drizzle and rub with olive oil and season with salt and freshly cracked black pepper.
Prepare your grill for indirect heat grilling. You can also add smoking woods to add aromatic wood smoke flavor.
Start grilling your BBQ turkey legs over indirect heat with your grill lid closed as much as possible. Begin basting with the basting mixture frequently to begin developing a rich, dark caramelized glaze.
Turn the turkey drumsticks periodically and continue basting frequently. Be sure to keep the lid closed between basting to retain heat and smoke.
Total cooking time will vary depending on your grill temperature. Check the doneness with a meat thermometer or by cutting one turkey leg open if need be.
When done, remove from the grill to a platter or plate and cover in foil to rest about 10 to 15 minutes.