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1. FOR THE PORK: Dissolve 1 cup salt, 1/2 cup sugar, and 3 tablespoons liquid smoke in 4 quarts cold water in large container. Submerge pork in brine, cover with plastic wrap, and refrigerate for 2 hours.
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2. While pork brines, combine mustard and remaining 2 teaspoons liquid smoke in small bowl; set aside. Combine black pepper, paprika, remaining 2 tablespoons sugar, remaining 2 teaspoons salt, and cayenne in second small bowl; set aside. Adjust oven rack to lower-middle position and heat oven to 325 degrees.
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3. Remove pork from brine and dry thoroughly with paper towels. Rub mustard mixture over entire surface of each piece of pork. Sprinkle entire surface of each piece with spice mixture. Place pork on wire rack set inside foil-lined rimmed baking sheet. Place piece of parchment paper over pork, then cover with sheet of aluminum foil, sealing edges to prevent moisture from escaping. Roast pork for 3 hours.
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4. Remove pork from oven; remove and discard foil and parchment. Carefully pour off liquid in bottom of baking sheet into fat separator and reserve for sauce. Return pork to oven and cook, uncovered, until well browned, tender, and internal temperature registers 200 degrees on instant-read thermometer, about 11/2 hours. Transfer pork to serving dish, tent loosely with foil, and let rest for 20 minutes.
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5. FOR THE SAUCE: While pork rests, pour 1/2 cup of defatted cooking liquid from fat separator into medium bowl; whisk in sauce ingredients.
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6. TO SERVE: Using 2 forks, shred pork into bite-sized pieces. Toss with 1 cup sauce and season with salt and pepper. Serve, passing remaining sauce separately.
