Mix the paprika, garlic power, brown sugar, dry mustard, and 3 T salt together in a small bowl. Rub the spice blend all over the pork and marinate, covered, in the refrigerator for at least 1 hour or up to overnight.
Preheat the oven to 300°. Put the pork in a roasting pan and bake for about 6 hoursuntil the inside temperature reaches 170° and the roast is falling apart.
Meanwhile, for barbecue sauce, combine the vinegar, mustard, ketchup, brown sugar, garlic, salt, cayenne, and pepper in a saucepan over medium heat. Simmer gently, stirring, for 10 minutes until the sugar dissolves.
Allow the meat to rest for about 10 minutes after removing from oven. While still warm, take 2 forks and "pull" the meat to form shreds.
Place the shredded pork in a bowl and pour 1/2 of the sauce on the shredded pork and mix well to coat.
Serve on hamburger buns with remaining sauce on the side.
Note: The above is Tyler Florence's actual recipe, but when I make it I make a few changes. It does not specify, but cover the roasting pan before putting in the oven. Also, pour a little water in the bottom of the roasting pan and check periodically to make sure it does not dry up completely. I found the drippings in the bottom of the pan burned if I didn't do this.
Note: For the sauce, I use 1C each of cider vinegar, mustard and ketchup as well as 1/2 C brown sugar and 1/2 t cayenne. I use all of the sauce to mix in the meat. I don't think half is enough. If you want more sauce on the side, I would suggest making more.