Light a grill. When the coals are covered with a light gray ash, push them to opposite sides of the grill and set a disposable drip pan in the center. If using a gas grill, turn off the center burners.
In a small bowl, mix 3 tablespoons of the salt with the paprika, brown sugar, garlic powder and 1 tablespoon of pepper. Sprinkle the mixture all over the ham, concentrating on the exposed part of the meat.
When the fire is medium-low (275° to 300°), toss 1 1/2 cups of the wood chips on the coals. (If using a gas grill, toss the chips into a smoker box.) Set the ham, skin side up, in the center of the hot grate above the drip pan and away from the coals. Cover the grill and cook the ham until darkly browned on the outside, cooked through and very tender inside, 7 to 8 hours. You'll need to replenish the coals every 1 to 2 hours; replenish the wood chips every hour for the first 4 hours. To test for doneness, use an instant-read meat thermometer: The internal temperature of the ham should be about 190°. (Yes, this is very well done—that's how you get meat tender enough to pull.) If the ham starts to brown too much, cover it loosely with aluminum foil for the last 2 hours or so, but remember that those browned bits are good, too.
Transfer the cooked ham to a cutting board. Cover loosely with aluminum foil and let rest for 20 minutes. Pull off any crisp skin and, if you're so inclined, finely chop it with a cleaver. (Southern pit masters call the crispy bits "brownies" and mix a little in with their pulled pork.)
Wearing heatproof food gloves, pull the meat off the bones in large pieces; discard any bones or lumps of fat. Using your fingers or 2 forks, pull each piece of pork into thin shreds, or finely chop with a cleaver. Transfer the pulled pork to a heatproof pan and stir in about 2 1/2 cups of the Vinegar Sauce, enough to keep the meat moist; add more as needed. Season the pulled pork with salt and pepper. If you are not quite ready to serve it, cover the pan with aluminum foil and place it on a warm (not hot) grill or in a low oven.
Lightly brush the cut sides of the hamburger buns with the melted butter. Lightly toast the buns on the grill. Just before serving, mound the pulled pork on the toasted buns, top with the Mustard Slaw and serve the remaining Vinegar Sauce on the side.
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