In a large stockpot or container, combine water, kosher salt, and sugar. Stir until completely dissolved.
Place turkey legs in the brine until completely submerged. Cover and refrigerate for about 4 to 6 hours.
Drain turkey from the brine, and rinse inside and out under cool running water. Pat dry with a paper towel.
Build a fire in the grill, leaving one side free of coals. When coals are covered with gray ash and the temperature is medium, they're ready.
Arrange the turkey legs in a single layer on the grill directly over the coals and cook, turning once every 5 minutes, for about 10 to 15 minutes or until skin begins to crisp and lightly char.
Transfer the turkey pieces to the cooler side of the grill, cover the grill, and continue to cook, turning occasionally, for about 40 to 45 minutes or until thermometer inserted in the thickest part of leg reads 165 F.
Remove from grill and serve hot.