Completely dissolve salt and sugar into 2 cups of hot water. Then, add the salt water solution to 10 cups of ice cold water in a large vessel. Completely submerge the chicken halves in the brine and place the vessel in the refrigerator for at least two hours. Overnight is fine if you have the time.
Light your grill and set up for two zone heating. Using a reliable leave in thermometer, stabilize the indirect heat zone at 425°F.
While grill is preheating, remove chicken from the brine and and pat it as dry as possible with paper towels.
Evenly coat the chicken halves with olive oil. Then season the chicken with dry rub, being sure to season under the wings.
Insert a leave in thermometer into the thickest portion of the breast and place the chicken on the grill over indirect heat. Close the lid and allow the chicken to cook until the internal temperature reaches 150°F. About 45 minutes.
Remove the chicken from the grill, tent with aluminum foil and allow the chicken the rest for ten minutes while carry over cooking continues to cook the chicken.