Combine paprika, pepper, brown sugar and salt in a small bowl, working out any clumps in the sugar with your fingers. Spread dry rub all over pork and rub in well. Allow to rest 15 minutes to 24 hours.
Prepare charcoal grill for indirect heat (place all the coals on one side). Soak 4 cups of wood chips in water for 15-30 minutes. Place the chips on the coals. Heat the top grill with the lid on for 5 minutes. Place the pork on the top grate on the side away from the coals. Open the top air vents fully, and position them over the meat. Open the bottom air vents half- to fully-open, depending on the weather (more closed on a windy day). Leave undisturbed 2 hours.
Transfer pork to a disposable aluminum roasting pan. Wrap with a double layer of aluminum foil. Place in a preheated 325°F oven for 2-3 hours longer, until fork-tender. Let rest covered for 30 minutes.
Remove the fat cap and set aside on a plate. Shred the meat into a bowl, discarding any large hunks of fat or gristle. Pull off the "bark" (the dark brown "skin") from the fat cap; discard the remaining fat.
Whisk the sauce ingredients together in a bowl. Pour 1/2 cup over the pulled pork; serve the rest on the side.
Serve between slices of white bread, with pickle slices.