Preheat oven to 350 degrees. Season pork all over with salt and pepper. Heat 2 tablespoons oil in a large, heavy-bottomed ovenproof pot over medium-high heat. Add pork and cook until browned on all sides, turning as needed, about 8 minutes total. Remove from pot.
Reduce heat to medium. Add remaining 1 tablespoon oil if pan is too dry, along with onion, garlic, caraway seeds, and a splash of water. Cook, stirring occasionally, until onion is translucent, 1 to 2 minutes. Add flour and cook, stirring, 30 seconds. Add vinegar and bring to a boil, then add cider and broth and return to a boil. Add cabbage, fennel, and 1 apple. Return to a boil, then reduce to a vigorous simmer, cover, and cook 7 minutes, stirring once.
Add pork, nestling it into vegetables. Transfer to oven and cook, uncovered, 15 minutes. Remove pork and add remaining 2 apples, turning vegetables and pushing apples down into liquid. Return pork to pot and continue cooking until a thermometer inserted in thickest part of pork registers 138 degrees, 20 to 25 minutes more. Remove pork and let rest at least 10 minutes before slicing. If necessary, return vegetables and apples to oven and cook, covered, until fork-tender, 10 to 20 minutes more.
Transfer vegetables and apples to a platter. Slice pork and add to platter. Stir butter into liquid in pot, then season with salt and vinegar. Serve pork and vegetables with sauce on the side.