Pre-heat oven to 350°F.
Season pork all over with salt and pepper.
In a large, heavy-bottomed ovenproof pot, over medium-high heat, heat 2 tablespoons oil.
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°F 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.
