Cut the pork shoulder into 2-inch chunks for even cooking throughout the braise. Heat 1 tablespoon of butter in a large, heavy-bottomed pot or Dutch oven over medium-high heat until it foams. Working in batches if necessary to avoid crowding, brown the pork chunks on all sides for 2-3 minutes per side, developing a deep golden crust that will add rich flavor to the final dish. Don't worry about cooking the meat through—you're just building flavor here. Remove the browned pork and set aside.

Return the browned pork to the pot and add the drained and rinsed sauerkraut, distributing it evenly over the meat. Cover the pot with a lid and reduce heat to low. Let the pork braise gently for about 2 hours, until the meat is completely tender and falls apart easily when pierced with a fork. The sauerkraut will create steam and acidity that tenderizes the pork while imparting classic flavor. I find that low and slow cooking really develops the complex, tangy flavor that makes this dish special—resisting the urge to turn up the heat makes all the difference.

While the pork braises (you can start this around the 1.5-hour mark), prepare your dumpling dough. In one bowl, whisk together the gluten-free flour, baking powder, salt, and xanthan gum—the xanthan gum is crucial here as it mimics gluten's binding properties and gives the dumplings structure. In a separate bowl, whisk together the room-temperature egg, whole milk, and melted cooled butter until well combined. This separation of wet and dry ingredients ensures even distribution and prevents overdevelopment of the dough. Pour the wet ingredients into the dry ingredients and gently stir with a spoon just until combined, adding a splash more milk if the dough is too thick to drop from a spoon. The dough should be thick and slightly lumpy, not smooth.

Once the pork is completely tender, carefully remove it from the pot with a slotted spoon and place on a cutting board. Shred or cut the meat into bite-sized pieces, discarding any large pieces of fat or bone. Return the shredded meat to the pot with the sauerkraut and increase the heat to bring the liquid to a gentle simmer. The meat will have released collagen and flavor into the braising liquid, creating a rich, flavorful base for the dumplings to cook in. Keep the pot at a consistent gentle simmer—this is the ideal environment for cooking the dumplings.

Using a spoon, drop golf ball–sized portions of the dumpling dough from Step 3 directly onto the simmering pork and sauerkraut, spacing them evenly so they have room to expand. Let them cook uncovered for 7 minutes—during this time, the bottom of each dumpling will set and begin to firm up. After 7 minutes, cover the pot with a lid and reduce heat to low, cooking for another 10 minutes. This two-stage method is important: the uncovered cooking sets the base of the dumplings, while the covered steaming finishes cooking the interior and keeps them light and fluffy. The dumplings are done when they've puffed up and a fork easily pierces the center without resistance.

Ladle the pork and sauerkraut into shallow bowls, making sure each serving gets several pieces of tender meat and some of the flavorful braising liquid. Top each bowl with 2-3 dumplings, spooning additional broth and sauerkraut around them. The dumplings will have absorbed some of the tangy sauerkraut flavor while remaining tender and satisfying. This is one of those dishes where everything improves as it cools slightly, so there's no need to rush to the table.