Add the brown sugar, smoked paprika, mustard powder, kosher salt, and pepper to a small bowl. Mix thoroughly.
Pat the pork shoulder dry with a paper towel. Trim the pork of any excess fat. Using your hands, coat the pork shoulder with the seasoning by rubbing the mixture into the meat. Then cut the meat into 6 pieces.
Set your Instant Pot's sauté function to high. Pour the vegetable oil into the Instant Pot. Once the oil is hot, place the pork into the pot. Work in batches to avoid crowding. Sear the meat on all sides, about 5 minutes per batch.
Turn the sauté function off and turn on the warm function. Using a pair of tongs, remove the pork from the Instant Pot and place it on a plate.
Pour the broth and apple cider vinegar into the pork drippings in the Instant Pot.
Using the tongs, place the pork back into the Instant Pot. Secure the lid and then close the sealing valve. Select the manual or pressure cook function and cook on high pressure for 1 hour.
When the pork is done, naturally release the pressure in the pot and carefully remove the lid.
Using tongs, transfer the pork to a large bowl, leaving the liquid in the pot.
Use 2 forks to shred the pork. Toss the shredded meat with your favorite barbecue sauce.