Pat pork roast dry with towel. Combine applesauce and seasonings into paste. Rub paste onto roast and refrigerate for at least one hour. Wrap entire roast in heavy-duty aluminum foil.
Set one grid at the highest setting and the others at the lowest setting. Set the burner to medium heat under the high grid. Place an oven-safe meat thermometer into the heart of the roast. Cook with lid closed or partially closed until the internal temperature is 170°F. Place apple wood chips on top of the flavor screen during last 30 minutes of cooking. Remove roast from grill, carve and serve.