Put the chicken in a large container or bowl and sprinkle generously with salt and pepper on both sides. Coat the chicken with the oil and sprinkle all over with the Cajun spice. Cover the container and refrigerate at least 2 hours and no more than 24 hours.
Make the coleslaw: Combine the cabbage, carrot and garlic in a large bowl and toss together. Add the oil, vinegar, sugar, and salt. Sprinkle with pepper and mix well. Refrigerate for at least one hour and ideally overnight. Before serving, mix well and add salt and pepper to taste.
Heat a grill, preferably charcoal. When the coals are red let the temperature fall back down to somewhere between 350 to 300 degrees F.
Place the chicken skin-side down on the rack of the grill and cover. You want to keep the smoke and flavor inside the grill. You also want your coals to burn slow and low, and by closing the lid the coals have less oxygen and burn slowly.
Check the color of the skin after 20 minutes. If the skin is nice and golden, flip the chicken; add more coals at the same time if the temperature has dropped below 275 degrees F. Keep cooking until the wings and the legs are easily removed from the body with 2 fingers, about 25 minutes.