Heat barbecue to medium-high. Place the chicken, oil and Cajun spice in a bowl and toss to coat the chicken. Season the chicken with salt, if needed. Turn the heat off on one side of the barbecue. Place chicken, skin-side up, on that side of barbecue. Close the lid and cook the chicken until cooked through, about 35 minutes, or until the temperature in the deepest part of the thigh registers 170 degrees F on an instant read meat thermometer. When done, to enhance its colour, set the chicken skin-side down on the hot side of the barbecue and cook 1 or 2 minutes more (do not let it burn).
Note: Cajun spice is sold in bottles in the spice aisle. If it contains salt, you won’t need to use any additional salt to season the chicken. If you want to make your own Cajun spice, see the expanded version of this recipe at thriftyfoods.com and scroll down to options.