In a large pot, combine the vegetable stock, kosher salt, brown sugar, peppercorns, rosemary, bay leaves, and garlic. Bring the mixture to a boil, stirring until the sugar and salt dissolve. Turn off the heat and let it cool completely.
Once the brine cools, add the iced water to the pot. Remove the chicken from the packaging and discard the neck and gizzard. Rinse the chicken in cold water. Submerge the whole chicken, breast-side down, in the brine, ensuring that the cavity of the chicken is filled with the brine solution. Add additional cold water if there is not enough brine to cover the chicken. Cover the pot and refrigerate for 4 hours.
After brining, remove the raw chicken from the brine, rinse it under cold water and pat it dry with paper towels.
Using butcher’s twine, truss the chicken by tying the legs together and tying the wings together behind the bird. This helps prevent the wings from burning during the cooking process.
Coat the entire chicken with olive oil. Then, sprinkle with kosher salt, freshly ground black pepper and paprika.
Setup your grill and rotisserie attachment. Slide the rotisserie rod down the center of the chicken. Carefully place the spit forks onto the rod. Slide them down until the tines are firmly inserted into the chicken to hold it in place. Tighten the thumb screws to secure the tines in place. Use a pair of pliers to gently tighten the thumb screws just past finger tight. Don’t make it too tight or the screw can break off.
Setup a Kamado Joe or Big Green Egg for direct heat at 350°F (the vent will be almost closed, open only about ¼”). Place the chicken into the rotisserie attachment directly over the hot coals, start the motor and close the lid. If you are using a rotisserie with a heating element to the rear, place a drip pan directly below the chicken to catch the fat drippings, which will make cleanup easier.
A 4-5 pound chicken will cook in about 1 hour and 10 minutes to 1 hour and 30 minutes. To check the internal temperature, insert an instant-read thermometer into the thickest part of the breast. Once your chicken reaches the 160°F, use heatproof gloves or hot pads to carefully remove it from the grill and place it on a cutting board with a juice trough. Tent the chicken with aluminum foil and let it rest for about 10 minutes. This will allow the juices to redistribute throughout the meat, giving you moist, tender and flavorful chicken.
After the chicken has rested, remove the rotisserie spit and rotisserie forks, then carve the chicken into pieces. Serve it with sides like grilled vegetables and garlic mashed potatoes. Use the leftovers for delicious meals like chicken Caesar salad or homemade chicken enchiladas.