Place a large aluminum pan under the side of the barbecue you will be cooking the turkey on. Heat the barbeque to high heat. Brush the turkey with the olive oil; sprinkle with barbecue spice. Tuck the turkey's wings under its body. Lightly oil the grill on one side of the barbecue. Set on the turkey, breast side up; turn off the burner directly underneath it. Close the lid and cook 75-95 minutes, or until the turkey in the deepest part of a thigh registers 185 degrees F on an instant read meat thermometer. While this occurs, combine glaze ingredients in a pot and bring to a simmer. Remove from the heat; baste the turkey with the glaze during the last few minutes of cooking.