Prepare the pork: Combine the mustard, honey, 1 teaspoon salt, 1 tablespoon pepper, the paprika and garlic in a bowl. Prick the pork on all sides (including the fat) with a paring knife, about 1/2 inch deep. Brush all over with the mustard mixture. Tightly wrap with plastic wrap and refrigerate at least 6 hours.
Unwrap the pork, place in a roasting pan and bring to room temperature. Meanwhile, preheat the oven to 250 degrees F.
Transfer the pork to the oven and pour 1 inch hot water into the pan. Cook 3 hours, adding water as needed to maintain 1 inch.
Remove the pan from the oven, add water if needed and cover tightly with foil. Return to the oven and cook 6 more hours. Turn off the oven and let the pork rest inside, about 1 hour.
Meanwhile, make the sauce: Melt the butter with the oil in a large skillet over medium heat. Add the onions, season with salt and pepper and cook, stirring occasionally, until tender, 6 to 8 minutes. Add the garlic and paprika and cook 2 to 3 more minutes (reduce the heat if the garlic starts to brown). Stir in the brown sugar, preserves, ketchup and chili sauce, stirring after each addition. Bring to a simmer, then stir in the vinegar. Add salt to taste.
Remove the pork from the oven and take the meat off the bone. Coarsely shred the meat using 2 carving forks. Toss with some of the barbecue sauce and pan juices and season with salt and pepper. Serve on rolls with coleslaw.
Photograph by Steve Giralt
Recipe courtesy of Sunny Anderson for Food Network Magazine