Preheat the oven to 325°. In a large Dutch oven or a heavy pot with a lid, heat the oil over medium-high. Season the pork all over with salt and pepper; add to the pot. Cook, turning occasionally, until browned all over, about 10 minutes. Transfer to a plate.
Reduce the heat to medium. Add the onion and rosemary. Cook, stirring often, until the onion is golden brown and softened, about 5 minutes. Add the cider to the pot and bring to a boil. Return the pork to the pot and cover.
Transfer the pot to the oven and cook until the meat is fork-tender and beginning to fall off the bone, 2 1/2 to 2 3/4 hours. Transfer the pork to a cutting board.
Spoon the fat from the sauce in the pot and discard along with the rosemary sprigs. Transfer remaining sauce to a blender and puree. Slice or coarsely shred the pork and serve with the sauce.
Try our Butternut & Brown Butter Polenta, also in photo.