Dissolve 1 cup salt, ½ cup brown sugar sugar and 3 tablespoon liquid smoke in 1 gallon (4 L) cold water in a large bowl. Place pork into brine, cover and refrigerate for at least 2 hours, up to overnight.
While pork is brining, stir together mustard and remaining 2 teaspoons liquid smoke in a small bowl. Mix pepper, paprika, cayenne and the remaining 2 Tablespoons brown sugar and 2 teaspoons salt together in a small bowl.
Adjust oven rack to lower middle position, heat oven to 325°F. Prepare a baking sheet by lining it with foil and set a wire rack over top.
Remove pork from brine, dry with paper towels. Brush mustard mixture over pork to cover. Sprinkle paprika mixture over pork and pat down to stick. Place pork on the wire rack over foil-covered baking sheet. Place a sheet of parchment paper over pork, then cover with foil, sealing the edges completely. Place baking sheet in oven and roast for 3 hours.
Remove pork from oven. Remove and discard the foil and parchment that are covering the pork. Carefully remove wire rack with the pork on top and set aside. Carefully pour pork drippings from foil-lined pan into a bowl or measuring cup and place in fridge to cool. Replace wire rack and pork and return to oven to cook for an additional 1 ½ hours, or until the center of the pork reads 200°F on an instant-read thermometer.
Transfer pork to a platter or cutting board and loosely cover with foil. Allow to rest for 20 minutes.