Cut the pork shoulder into large 3-inch chunks and pat dry. Season with salt and pepper. Heat oil in a cast iron Dutch oven over medium-high heat. Sear the pork chunks for 3-4 minutes per side until a deep mahogany crust forms. Remove the pork and set aside.

Lower heat to medium. In the remaining fat, sauté the chopped onion for 5 minutes until translucent. Add the minced garlic and caraway seeds, cooking for 1 more minute until fragrant.

Pour the beer or apple cider into the pot. Vigorously scrape up the browned bits (fond) from the bottom using a wooden spoon. Let the liquid reduce by half.

Stir the slightly drained sauerkraut and broth into the pot. Return the pork chunks to the pot, nestling them into the kraut. Bring to a simmer, cover tightly, and transfer to a 325°F (160°C) oven. Braise for 2.5 to 3 hours until the meat is fork-tender.

Remove from the oven. Using two forks, shred the pork directly in the pot. Toss the meat with the hot sauerkraut and juices. Top generously with fresh chopped parsley before serving.