Remove the neck and giblets from the turkey cavity and reserve for another use. Lightly brush or spray the turkey all over with oil. Season generously with salt and pepper inside and out.
In a medium saucepan over medium-high heat, cook the chicken stock, butter, marjoram, thyme, garlic, and 1/2 teaspoon of pepper until the butter has melted. Pour 1 cup of the chicken stock mixture into a small bowl. Draw the mixture in the bowl into a kitchen syringe. Inject the syringe into the drumsticks, thighs, and breast of the turkey, refilling the syringe each time. Pour the remaining 2 cups of the chicken stock mixture inside a heavy-duty roasting pan. Place the turkey, breast side up, on a roasting rack, and set inside the pan.
Follow the grill's instructions for using wood chips. Grill the turkey over Indirect Medium heat, using wood chips for the first 30 minutes. Check the turkey after the first hour. If any parts are getting too dark, wrap them tightly with aluminum foil. Check again after another hour and cover any dark areas with foil. The turkey is done when the internal temperature reaches 165°F in the breast and 175°F in the thickest part of the thigh, 2 1/4 to 3 hours. Transfer the turkey to a cutting board and let rest for 20 to 30 minutes before carving (the internal temperatures will rise 5°F to 10°F during resting). Serve warm or at room temperature.