Pat the pork shoulder dry with paper towels. Season generously on all sides with salt and black pepper.

Heat the vegetable oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Carefully place the seasoned pork in the pot and sear on all sides until a deep golden-brown crust forms, about 3-4 minutes per side. Remove the pork from the pot and set it aside on a plate.

Reduce the heat to medium. Add the sliced onion and apple to the pot, stirring to coat with the pork drippings. Cook until softened, about 5-7 minutes. Add the minced garlic and caraway seeds and cook for one more minute until fragrant.

Pour in the beer or chicken broth. Use a wooden spoon to scrape up any browned bits from the bottom of the pot. Bring the liquid to a simmer.

Turn off the heat. Spread half of the sauerkraut in an even layer. Place the seared pork on top. Cover the pork with the remaining sauerkraut. Pour any juices from the plate with the pork back into the pot.

Bring the pot to a simmer over medium heat. Once simmering, reduce the heat to low, cover the pot with a tight-fitting lid, and let it cook for 2.5 to 3 hours. The pork should be very tender and easily shred with a fork.

Remove the pot from the heat and let the pork rest in the juices for 10-15 minutes. You can serve the pork sliced or shredded, with a generous helping of the sauerkraut and juices.