Heat a tablespoon or two of oil in a skillet (if you'll be using a slow cooker) or a Dutch oven over medium-high heat and brown the shoulder or loin on all sides. If using kielbasa, cut the sausages in half lengthwise, then crosswise into 1-inch-wide half-moons. No need to brown them, just put in the Dutch oven or slow cooker. Add the sauerkraut, including the brine, onion and apple to the meat in the Dutch oven. You may need to add up to 2 cups water if your kraut is fairly dry. Bring to a boil. Lower to a simmer and cook low and slow for 3 to 4 hours, checking occasionally and adding water if it starts to dry out.
If using a slow cooker, combine the meat, sauerkraut and brine, onion and apple in the cooker, cover, and cook on high for 6 hours. (You shouldn't have to add any liquid.)
When done, the kielbasa will have darkened significantly and the pork will be fork-tender and falling off the bone or separating from the fat. Using two forks, shred the pork into bite-size pieces and discard any bones and large chunks of fat.
Serve in bowls over mashed potatoes, or with applesauce on the side, if desired.
Recipe courtesy Casey Barber, author of the forthcoming Classic Snacks Made from Scratch: 70 Homemade Versions of Your Favorite Brand-Name Treats (Ulysses Press)