Season the pork butt with salt, pepper and the smoked paprika.
Add the meat to a small Dutch oven along with the BBQ sauce, apple juice, vinegar, hot sauce, Worcestershire and soy sauce. Cover the pot tightly with its lid and put it in the oven until the pork is extremely tender, about 2 hours. Remove the pan from the oven and let it cool slightly before shredding. Shred the meat in the Dutch oven, using 2 forks, so the meat can absorb more of the flavors of the cooking liquid. Let cool.
For the dipping sauce: While the pork cooks, mix all the dipping sauce ingredients together in a small bowl. The sauce can be stored in the refrigerator for up to 3 days in a tightly covered container.
For the egg rolls:
In a deep-fryer or heavy-bottomed pot, heat enough peanut oil to come halfway up the sides of the pot, to 350 degrees F.
Beat the egg and water together in a small bowl. Reserve.
Unwrap the egg roll wrappers. Brush the edges with the egg wash and set them on your work surface with the triangle edges pointing towards you. Divide some of the pork among the lower third of the wrappers. Top the pork evenly with the coleslaw. Fold the bottom portion of the wrapper over the filling. Fold both the edges inwards and roll the wrapper up completely. Press on the open flaps to firmly secure the rolls, then set them aside, seam side down.
Add the egg rolls, in batches, to the deep-fryer and fry, flipping with tongs, until they are crisp and golden brown, about 2 minutes. Remove them from deep-fryer and let drain on a paper towel-lined sheet tray. Transfer the rolls to a serving platter and serve hot along with the peach dipping sauce.