Mix the paprika, garlic power, brown sugar, dry mustard, and salt together in a small bowl. Rub the spice blend all over the pork and marinate for as long as you have time for, as little as 1 hour or up to overnight, covered, in the refrigerator.
Preheat the oven to 300 degrees F.
Put the pork in a roasting pan and bake for about 6 hours. Basically, roast the pork until it's falling apart and an instant-read thermometer inserted into the thickest part registers 170 degrees F.
To make the barbecue sauce: combine the vinegar, mustard, ketchup, brown sugar, garlic, salt, cayenne, and black pepper in a saucepan over medium heat. Simmer gently, stirring, for 10 minutes until the sugar dissolves.
Remove the pork roast from the oven and transfer to a large platter. Allow the meat to rest for about 10 minutes. While still warm, take 2 forks and "pull" the meat to form shreds. Using 2 forks, shred the pork by steadying the meat with 1 fork and pulling it away with the other. Put the shredded pork in a bowl. Pour 1/2 of the sauce on the shredded pork and mix well to coat.
To serve, spoon the pulled pork mixture onto the bottom 1/2 of the hamburger bun, and top with the spicy slaw. Serve with pickle spears and the remaining sauce on the side.