Cut and season the pork: Cut the pork shoulder into 3–4 large chunks for even cooking. In a small bowl, mix salt, pepper, smoked paprika, garlic powder, onion powder, ground mustard, cayenne, and brown sugar.
Rub the mixture all over the pork pieces until well coated.
Sauté to build flavor: Set the Instant Pot to Sauté (High). Add oil. When hot, sear pork in batches for 2–3 minutes per side until browned.
Transfer to a plate as you go. Don’t overcrowd the pot.
Deglaze the pot: Add a splash of broth and scrape up browned bits with a wooden spoon. This prevents a burn notice and adds flavor.
Load the pot: Add sliced onion and smashed garlic to the bottom.
Pour in the remaining broth and the apple cider vinegar. Return the pork and any juices to the pot. Add 1/2 cup BBQ sauce and gently stir to coat.
Pressure cook: Lock the lid and set the valve to Sealing.
Cook on High Pressure for 60–70 minutes (60 for 3 lb, 70 for 4 lb), then allow a 15–20 minute natural release. Manually release any remaining pressure.
Shred the pork: Transfer pork to a large bowl. Skim excess fat from the cooking liquid if desired.
Shred with two forks until the meat is in bite-size strands.
Reduce and sauce: Set the Instant Pot to Sauté. Let the cooking liquid simmer for 8–10 minutes to reduce by about one-third. Stir in 1/2 to 1 cup additional BBQ sauce, adjusting to taste.
Return the shredded pork to the pot and toss to coat. If it looks dry, add a splash of broth or more sauce.
Serve: Taste and adjust seasoning with salt, pepper, or a splash of vinegar for brightness. Pile onto buns with coleslaw or serve over rice, baked potatoes, or in tacos.