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Trim most of the visible fat from the outside of the pork butt leaving about 1/8- to 1/4-inch layer of fat on top. Place pork in baking pan. Rub all over with Applewood Rub. Cover with plastic wrap. Let stand at room temperature 1 hour.
Prepare grill for indirect medium-low heat (250°F to 325°F). Preheat grill by turning all burners to high with lid closed. Turn off burner(s) on one side of the grill. Turn burner on the other side to medium-low. Carefully place foil pan under grates on unlit side of grill to catch drippings. Place pork, fat-side up, on unlit side of grill. Close lid.
Note: To maintain medium-low heat (250°F to 325°F), keep lid closed and adjust lit burner as necessary. Directions were developed using a gas grill. Grills vary; cooking time is approximate.
Grill about 6 hours or until internal temperature of thickest part of pork butt is 165°F. Carefully remove pork from grill and wrap in foil.
Return pork to grill, placing it back on unlit side. Grill 1 to 2 hours longer or until internal temperature is 195°F to 200°F and meat is tender and bone twists easily.
Remove pork from grill. Let stand 15 to 20 minutes. Slice or shred with 2 forks. Serve with warm pan juices or your favorite barbecue sauce, if desired.