Remove the baking dish from the oven and discard the foil. Transfer the pork to a cutting board (reserve the braising liquid). Let the pork cool slightly before removing skin and shredding. Use tongs to remove and discard the skin and fat from the meat. Gently pull the pork apart into smaller pieces, then transfer them to a rimmed heat-safe serving platter. Cover the serving platter loosely with foil and then place in the oven to keep warm while making the sauce.
Use a spoon to skim off any fat from the top of the braising liquid, then strain the remaining defatted liquid through a fine-mesh sieve set over a small saucepan (discard the onion, garlic, thyme, and bay leaves). Simmer over medium-high heat until slightly reduced, about 10 minutes. Remove the pork from the oven, discard the foil, pour the sauce over the pork, and serve.
Variation: Quick Pulled Pork
Cook the pork and reduce the braising liquid as instructed, then stir in 3 cups of storebought barbecue sauce and continue to cook until hot. Use 2 forks to shred the pork. Stir the shredded pork into the sauce and cook until warmed through. Place the pork on potato rolls. Serve with coleslaw and bread and butter pickles.
Variation: Pork Enchiladas
Cook the pork and skim and strain the braising liquid as instructed. In a large saucepan, combine ¾ cup of the braising liquid (freeze the rest to use another time) with three (15-ounce) cans of your favorite enchilada sauce. Simmer over medium-high heat until the sauce has thickened slightly, about 10 minutes. Remove 2 cups of the sauce from the pan and set it aside. Use 2 forks to shred the pork and add 3 cups of the shredded pork to the enchilada sauce in the pan and toss to combine (save the rest of the pork to use another time). Place eight 6-inch corn tortillas on a work surface, spread one eighth of the shredded pork down the center of each tortilla, and roll. Place the enchiladas seam side down in a lightly greased 9-by-13-inch baking dish and pour the reserved enchilada sauce evenly over the top. Cover with shredded pepper jack or cheddar cheese and bake at CONVECTION/BAKE 350°F until golden brown and bubbling, about 30 minutes. Let stand for 10 minutes. Serve with sour cream, chopped fresh cilantro, and lime wedges.