Combine the salt, pepper, smoked paprika, and dried thyme and rub the mixture all over the pork shoulder.
Heat the oil in a large skillet over medium-high heat. Sear the pork on all sides, about 4 minutes per side, until well browned. Transfer to the slow cooker, fat cap up.
Scatter the onion slices, apple wedges, and smashed garlic around and over the pork.
Pour the cider into the still-hot skillet and scrape up the browned bits. Whisk in the chicken broth, cider vinegar, mustard, and brown sugar, then pour the mixture around the pork so the rub stays on the meat.
Tuck in the bay leaves and thyme sprigs. Cover and cook on LOW for 8 hours, until the pork pulls apart with gentle pressure from a fork.
Lift the pork onto a cutting board and tent with foil. Discard the bay leaves and thyme stems.
Skim the fat from the cooking liquid. Mash the softened apples and onions into the liquid with a potato masher, or blend smooth with an immersion blender.
Pour the sauce into a saucepan and simmer for 8-10 minutes to reduce. Off the heat, whisk in the cold butter a piece at a time until glossy. Season with salt and pepper.
Pull the pork into large pieces, discarding the bone and large pockets of fat. Return it to the sauce or spoon the sauce over the top.
Serve over mashed potatoes or buttered noodles, finished with parsley.