Cut the pork shoulder into 2-inch chunks and season generously with salt and freshly ground black pepper on all sides—this creates a flavorful crust when seared. Dice the onion into 1/2-inch pieces and have all ingredients measured and ready before you begin cooking. Preheat your oven to 300°F so it reaches temperature while you're building the braising liquid.

Heat the olive oil in a large, heavy-bottomed pot or Dutch oven over medium-high heat until it shimmers. Working in batches if needed to avoid crowding, sear the seasoned pork chunks for 3-4 minutes per side until deeply browned on all surfaces. I always let the pork sit undisturbed for a minute or two on each side—this creates a proper crust instead of steaming the meat. Transfer the browned pork to a plate and set aside; don't worry about it being cooked through, the oven will finish it.

In the same pot with the rendered pork fat, add the diced onion and cook for 3-4 minutes, stirring occasionally, until softened and beginning to turn golden. Sprinkle the paprika over the onions and stir constantly for about 30 seconds to bloom the spice and release its flavor. Pour in the dark beer, scraping up any flavorful browned bits stuck to the bottom of the pot with a wooden spoon—this is called deglazing and adds tremendous depth to your braising liquid.

Add the water and bay leaves to the pot and bring the liquid to a gentle simmer. Return the browned pork to the pot, nestling it into the liquid so it's mostly submerged. Cover the pot with a lid and transfer it to the preheated 300°F oven. Braise for 1 to 1.5 hours until the pork is completely tender and easily breaks apart with a fork—the low, slow heat ensures the meat becomes melt-in-your-mouth tender rather than tough and dry.

Remove the pot from the oven and arrange the drained sauerkraut around and on top of the pork. If the liquid seems too reduced or the mixture looks dry, add a splash of water (about 1/4 to 1/2 cup) to keep everything moist. I prefer the sauerkraut to have just enough braising liquid to keep it from sticking without making it soupy. Cover the pot again and return it to the oven for a final 20 minutes, allowing the sauerkraut to warm through and absorb the flavors from the pork and braising liquid.

Remove the pot from the oven and let it rest, covered, for 5 minutes before serving. This allows the flavors to settle and any excess heat to distribute evenly. Taste the braising liquid and adjust the seasoning with additional salt and pepper as needed. Serve the tender pork and sauerkraut directly from the pot or transfer to a serving dish, making sure to include plenty of the flavorful braising liquid with each portion.