Make the pork:
Liberally rub pork shoulder with salt and pepper. Cut 2 garlic cloves into 16 slivers. Make sixteen 1-inch incisions and insert a sliver in each. Cover and refrigerate for 6 to 12 hours.
In a Dutch oven or large ovenproof casserole, heat 2 tablespoons of oil over medium-high heat.Add pork and brown on all sides. Transfer pork to a platter. Pour out all but 2 tablespoon of fat and return Dutch oven to medium-high heat. Chop 2 more garlic cloves.Add carrot, celery, onion, and chopped garlic and cook, stirring occasionally, until softened, about 3 minutes. Stir in 1 tablespoon of anchovies and 1 1/2 teaspoons chili flakes and stir until fragrant, about 1 minute. Add tomatoes with their juices, thyme, and bay leaf to Dutch oven and bring to a boil, scraping up browned bits on the bottom with a wooden spoon. Return pork with any juices to the Dutch oven.Add enough hot water to almost cover the pork and return to a boil.
Tightly cover Dutch oven and bake in 375 degrees F. oven, turning pork occasionally, adding hot water as needed if liquid reduces, until pork is very tender, about 3 hours. Remove from oven and let pork cool in sauce until easy to handle, 1 to 2 hours. Discard the thyme branch and bay leaf.
Transfer pork to a carving board and shred with fingers or two forks. Set aside half of pulled pork for another use (see note below). Skim off fat from surface of sauce. Stir half of the pork into the sauce. Keep warm over low heat.
In large sauté pan, preferably nonstick, heat the remaining 2 tablespoons of oil over medium heat. Chop the remaining garlic. Add the garlic with the capers and remaining 1 tablespoon anchovies, and 1/2 teaspoon chili flakes and cook, stirring often, until garlic is lightly browned, about 1 minute. Add the cherry tomatoes and olives and cook, stirring occasionally, until tomatoes soften, about 2 minutes. Set aside.
Cook the pasta:
Boil pappardelle in a large pot of salted water until al dente, according to box directions. Scoop out and reserve 1/2 cup of the pasta cooking water. Drain well and return to the pot. Add sauce, cherry tomato mixture, and half of the basil, and the Romano cheese. Mix well. If the sauce seems thick, stir in some of the reserved pasta water.
Return the sauté pan to medium heat. Add the reserved pasta and cook until hot, about 2 minutes. Serve the pasta in bowls, topped with the pork and remaining basil. Pass additional oil for drizzling and with more Romano shaved from a chunk of cheese with a vegetable peeler, if desired.
Note: The recipe will yield about 2 pounds pulled pork. Use half of the pulled pork for this sauce, and reserve the remainder for another meal. Freeze the pork in a tightly covered freezer-safe container for up to 3 months; thaw and reheat before serving.It makes a great sandwich on a crusty roll, topped with melted provolone and filled with sautéed broccoli rabe and garlic.
Recipe courtesy of Jose Mendin at Pubbelly Noodle Bar, Miami FL