Trim most of the visible fat from the chops. If the chops are wet, pat them dry with paper towels and sprinkle with salt and pepper to taste.
Put the flour in a bag, add the chops a few at a time, shake well. As you take the chops out, try to shake off as much flour as possible.
Over medium heat, heat enough olive oil in the bottom of a skillet to just cover the bottom of the pan. Brown the chops in batches, adding more oil a tablespoon at a time if needed to prevent sticking and burning.
Dump the sauerkraut in a large bowl, add the brown sugar and caraway seeds and mix well. Use more or less brown sugar depending how sweet or tart you like sauerkraut. Lightly oil a large roasting pan. I used the roasting pan that came with my over - minus the rack. Evenly distribute the sauerkraut in the bottom of the pan.
Quarter the potatoes, slice the apple. I left the peels on both because that's what we like - but peel them if you prefer. Place these over the sauerkraut.
As the chops are browned, place them on top of the sauerkraut, potato, apple mixture. Pour the apple cider over all. Cover tightly with aluminum foil and bake 45 minutes at 350 degrees.
Remove the aluminum foil and bake 15 - 20 more minutes until the chops are tender and the whole dish is nicely browned on top.