In a large pot combine the brine ingredients. Stir well to dissolve the sugar and salt.
Submerge the chicken in the brine, breast side down, and refrigerate for 6 to 12 hours.
Prepare the grill for indirect cooking over medium heat (350° to 450°F).
Remove the chicken from the pot and discard the brine. Pat the chicken dry with paper towels. Truss the chicken with butcher’s twine.
Following the grill’s instructions, secure the chicken in the middle of a rotisserie spit, put the spit in place, and turn on the motor. Place a large disposable foil pan underneath the chicken to catch the drippings. Cook the chicken over indirect medium heat, with the lid closed, for 1 hour.
If your grill has an infrared burner at the back of the grill, after 1 hour of cooking, light that burner and set it to medium heat (leaving the regular outside burners on medium and the middle burners turned off). If your grill does not have an infrared burner, continue to cook the chicken as you were doing. Either way, cook the chicken until the surface is deep golden brown and the internal temperature reaches 160° to 165°F in the thickest part of the thigh (not touching the bone). This should take 10 to 20 minutes with the infrared burner and 20 to 30 minutes without the infrared burner. Watch carefully that the chicken skin does not burn.
When the chicken is fully cooked, turn off the rotisserie motor and remove the spit from the grill. Tilt the chicken upright over the foil pan so that the liquid that has accumulated in the chicken’s cavity pours into the pan. Let rest for 10 to 15 minutes (the internal temperature will rise 5 to 10 degrees during this time). Transfer the chicken from the spit to a cutting board. Cut into serving pieces. Serve warm.