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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.