Preheat oven to 300°F (150°C). Cut the pork shoulder into 2 large chunks, pat dry, rub with the kosher salt, black pepper, oregano, and cumin, then sear in 2 tbsp vegetable oil over medium-high heat, about 4 minutes per side, until deep brown on all surfaces.

Remove the pork to a plate. In the same pot over medium heat, cook the 4 smashed garlic cloves and 1/2 chopped white onion for 2 minutes. Pour in the 1 cup water, add the 2 bay leaves and 1 tsp crushed juniper berries, and scrape up all the browned bits from the bottom.

Return the pork and any resting juices to the pot. The liquid should reach about one-third up the sides of the pork. Cover and braise in the oven at 300°F (150°C) for 2.5 to 3 hours, until the pork pulls apart easily with two forks.

Transfer the pork to a large rimmed baking sheet and discard the bay leaves and juniper berries. Shred the pork into varied chunks with 2 forks, pour 2 to 3 tbsp of the braising liquid over the top, and spread into a single layer.

Broil on high with the rack 6 inches from the element for 3 to 5 minutes, toss, and broil for another 2 to 3 minutes as needed until the edges are dark and crispy. Transfer to a warm serving platter, spoon over any remaining braising juices, scatter the fresh cilantro on top, and serve with lime wedges.