Trim and cut the meat: Trim the thick fat cap on the meat to ⅛ inch thick, leaving the rest of the fat. (Yes, it is very fatty… don’t worry, we are going to cook the extra fat off later). Remove the bone from the meat if needed, then cut the meat into 2-inch chunks.
Add to Dutch Oven: Add the pork, water, onion, lime juice, orange juice, oregano, cumin, bay leaves, salt, and pepper to a 6.5 qt or larger dutch oven. Add the spent orange halves on top.
Simmer: Place the pot on the stove over medium-high heat and bring to a simmer, stirring occasionally.
Cook in the oven: Once simmering, cover pot with lid and transfer to the oven. Cook for about 2 hours, or until the meat is tender and falls apart when prodded with a fork. At the 1 hour mark remove the pot from the oven and flip the pieces of meat, bringing the pieces from the bottom to the top and vice versa.
Reduce the sauce: Once tender, remove the meat from the pot and onto a cutting board. Discard the bay leaves. Then place the pot with the remaining juices on the stove over medium-high heat. Bring to a rapid simmer and cook, stirring often, until the sauce is thickened, syrupy, and a rubber spatula leaves a trail when dragged across the bottom of the pot (about 15 minutes). You should have about 1 cup of liquid remaining.
Shred the meat: While the sauce is reducing, use two forks to shred each chunk into 3 pieces (2 pieces if the chunk is small).
Toss meat and liquids: Toss the shredded meat with the reduced liquids until the meat is well coated. Season with additional salt and pepper if needed.
Broil: Line a sheet pan with foil, then place a wire rack on top of the sheet pan. Line the meat across the wire rack (it will take up most of the rack). Then place on the middle rack in the oven and broil for 4-7 minutes, or until the edges are golden and crispy but not burned. Flip the meat and broil again for 4-7 minutes.
Serve: Serve while hot with warm tortillas and toppings. Enjoy!