Cut the pork into large pieces. You will need to cut around the bone. Discard the bone.
In a small bowl, combine smoked paprika, chili powder, cumin, garlic powder, onion powder, ground mustard, salt, pepper, and ½ tablespoon brown sugar.
Sprinkle the seasoning mixture over all sides of the pork. Rub to fully coat the pork in seasonings.
Set instant pot to the “sauté” setting. When warm, add the olive oil. Let olive oil heat for about 30 seconds, then add a few large pieces of pork. Sear for a few minutes per side, until browned on all sides. Remove pieces and continue with the rest. Do not overcrowd the pot, or else the pork won't brown.
Cancel the sauté setting. Carefully pour water and applesauce into the pot, scraping the bottom to remove any brown bits.* Once brown bits have released from the bottom, add the ketchup, ½ cup brown sugar, apple cider vinegar, and worcestershire sauce. Stir to combine.
Carefully add pork back into the pot. Cover instant pot with the lid, set valve to the “sealing” position and pressure cook on high for 60 minutes.
When the timer is up, let pressure release naturally for 20 minutes before switching the valve to the “venting” position and manually releasing any remaining pressure.
Remove pork to a large cutting board to shred with two forks. It should be fall-apart tender.
While shredding the pork, place instant pot back on the “sauté” setting to simmer the sauce and thicken it. Simmer for 30 minutes, or until reduced and starting to become thick like BBQ sauce. Stir occasionally so the bottom doesn’t burn.
Add pork back into the pot and stir to coat in sauce.
Serve warm on hamburger buns with coleslaw and pickles, on top of nachos, or on top of a baked potato.