Place the turkey breast side-down on a cutting board. With a large knife or heavy-duty kitchen shears, carefully cut along either side of the back bone. Remove backbone and save for stock or discard. Spread turkey open, and then cut in half down the middle of the breast bone. Freeze 1 of the turkey halves for another meal. Preheat the barbecue to medium heat, and then brush the other turkey half on both side sides with the olive oil.
Combine the spices and herbs and rub all over the turkey. Grill the turkey breast side up, 5 minutes. Turn over and cook on the skin side until nicely coloured, do not burn. Turn the heat off on one side of the barbecue. Move the turkey to that side. (Heat from the other side will rise, circulate, and cook the bird). Close the lid and cook, turning occasionally, 75 minutes, or until internal temperature reaches 180°F on a meat thermometer. Rest 10 minutes before carving. Baste the turkey with your favourite barbecue near the end of cooking, if desired.