If using a gas grill, preheat to high on one side; put soaked wood chips in a smoker box. Once smoking, reduce the heat to maintain a temperature of 275 degrees F and cook the pork, covered, on the cooler side of the grill.
Make the Neelys' go-to barbecue seasoning: Mix the paprika, sugar and onion powder in a bowl. Transfer 3 tablespoons seasoning to a separate bowl, add 2 tablespoons salt and 3 tablespoons pepper, and massage onto the pork. Cover with plastic wrap and refrigerate at least 2 hours or up to 1 day. (Reserve the remaining barbecue seasoning.)
Prepare the wood chips: Soak 6 cups wood chips in water, about 15 minutes, then drain. Don't oversoak, or the wood will snuff out the fire.
Light the grill: Fill a smoker or kettle grill with charcoal and light. (Pat uses lighter fluid; you can also use a chimney starter.) When the coals are mostly white, spread them out with tongs. Spread 1/2 cup of the wood chips over the coals (use 1 cup for a kettle grill). The temperature of the grill should be about 275 degrees F.
Cook the pork: Place the pork fat-side down on a rack in the smoker or on the grill. Cover and cook, rotating the pork every hour or so, until a thermometer inserted into the center registers 165 degrees F, about 6 hours total.
Feed the grill: As the pork cooks, add more charcoal and wood chips to keep the temperature between 250 degrees F and 275 degrees F and to maintain the smoke level.
Make the sauce: Meanwhile, mix the ketchup, 1 cup water, both sugars, 1 1/2 teaspoons pepper, the onion and mustard powders, lemon juice, Worcestershire sauce, vinegar, corn syrup and 1 tablespoon of the reserved barbecue seasoning in a saucepan over high heat. Bring to a boil, stirring, then reduce the heat to low and simmer, uncovered, stirring occasionally, at least 2 hours. Let cool, then reheat on the grill when ready to use.
Shred the pork: Transfer the pork to a rimmed baking sheet (you'll want to catch all the flavorful juices) and let stand until cool enough to handle. Shred into bite-size pieces, pile on a platter and pour any juices from the baking sheet on top.
Make the sandwiches: Mound the pork on bun bottoms, paint with a little barbecue sauce, top with slaw and cover with the bun tops. The best sandwich ever!