Remove the pork loin from refrigeration and let it sit at room temperature for 15-20 minutes—this ensures even cooking throughout the meat. While it warms, prepare all your ingredients: drain and rinse the sauerkraut, slice the apples into 1/4-inch thick pieces, dice the onion into 1/2-inch pieces, and mince the garlic. Pat the pork dry with paper towels and generously season all sides with salt and pepper. Heat the oil in a large Dutch oven over medium-high heat until it shimmers (about 2 minutes). Once hot, sear the pork for 4 minutes per side until deeply browned on all surfaces—this creates a flavorful crust that locks in juices. Remove the pork and set aside on a plate.
In the same Dutch oven with the browned bits still clinging to the bottom, add the diced onion and sauté for 3-4 minutes until softened and just beginning to turn golden. Add the minced garlic and cook for 30 seconds until fragrant—this brief cooking mellows the garlic's raw bite while infusing the oil. Stir in the drained sauerkraut, apple slices, brown sugar, chicken soup base, water, and a pinch of salt and pepper. Mix everything together well, scraping the bottom of the pot to incorporate all those flavorful browned bits from searing the pork. I find that combining all the kraut ingredients together first helps the flavors meld before the pork returns to the pot.
Nestle the browned pork loin back into the sauerkraut mixture, making sure it's partially submerged in the liquid. Cover the Dutch oven with its lid and transfer it to a 325°F oven. Braise for 1.5 hours until the pork is fork-tender and the internal temperature reaches 145-150°F. While the pork braises, you'll have time to cook the noodles in the next step.
About 30 minutes before the pork finishes braising, bring a large pot of salted water to a rolling boil. Add the egg noodles and cook according to package directions (usually 8-10 minutes) until they're al dente—tender but still with a slight bite. Drain the noodles well and immediately toss them with the butter while they're still hot, which helps coat them evenly and prevents sticking. Set the noodles aside and keep them warm.
Remove the Dutch oven from the oven and carefully slice or cut the pork into serving-sized pieces. Taste the sauerkraut mixture and adjust the seasoning with salt and pepper as needed—I like to add a pinch more salt to brighten all the flavors. Divide the buttered noodles among serving bowls or plates, then top each portion with sliced pork and generous ladlefuls of the sauerkraut mixture along with its braising liquid. Finish with a sprinkle of fresh chopped parsley for color and a fresh herbal note that cuts through the rich, savory flavors.