- Lay the pork flat, boned side up, on a cutting board. Using a sharp knife, trim away any large pockets of fat. Starting at the thinner end of the meat, roll up the pork and securely tie the roll at regular intervals with kitchen twine. Season the pork generously with salt and pepper.
- In a fry pan over medium-high heat, melt the butter with the canola oil. Add the pork and cook, turning frequently, until browned on all sides, about 10 minutes. Transfer the pork to a platter.
- Return the pan to medium-high heat, add the onion, apples and thyme, and sauté until the onion and apples are lightly browned, about 5 minutes. Transfer the apple mixture to a bowl.
- *Oven method:* Preheat an oven to 325°F. Cover the bottom of a large Dutch oven with the sauerkraut. Sprinkle with the brown sugar and caraway seeds. Place the pork on top and surround with the apple mixture. Pour in the wine mixture. Cover, transfer to the oven and cook until the pork is fork-tender and shreds easily, 4 to 5 hours.
- *Slow-cooker method:* Cover the bottom of a slow cooker with the sauerkraut. Sprinkle with the brown sugar and caraway seeds. Place the pork on top and surround with the apple mixture. Pour in the wine mixture. Cover and cook on low according to the manufacturer's instructions until the pork is fork-tender and shreds easily, 8 to 10 hours.
- Transfer the pork to a carving board. Using a sharp knife, cut the pork crosswise into slices about 1/2 inch thick, removing the twine as you slice. Place the sauerkraut on a large platter and top with the pork slices. Surround with the apples and serve immediately.