Blend sage, allspice, coriander, nutmeg, pepper, and seasoning salt in food processor. Pat pork roast dry with towel. By hand, press spice mix into the fat of roast. Wrap entire roast in heavy-duty aluminum foil.
Set one grid at the highest setting and the other 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. As the roast approaches the internal temperature, open the foil and coat the top of the roast with a mixture of the applesauce and brown sugar. Place wood chips on top of the flavor screen during last 30 minutes of cooking. Remove roast from grill, carve and serve.