Press the “Sauté” button on the Instant Pot. Once the pot is hot, add 1 tablespoon olive oil and heat.
Add half the pork shoulder and season with 1/2 teaspoon garlic powder, plus salt and black pepper, to taste. Sear until pork is browned on all sides, approximately 1-2 minutes per side, before transferring to a large, rimmed plate. Set aside.
Repeat with another tablespoon of olive oil and the remaining pork and garlic powder. Season with additional salt and black pepper, as desired. Once seared, transfer to the rimmed plate with the rest of the pork and set aside.
Add remaining olive oil, plus the red onion and shallot. Cook, stirring frequently, until soft, approximately 3-4 minutes.
Deglaze Instant Pot by adding the chicken broth and gently scraping the bottom of the container with a wooden spoon to loosen any browned bits.
Pour in the apple cider vinegar and 3/4 cups of barbecue sauce. Return the seared pork and its juices to the unit and cover, locking the lid into place.
Set the pressure valve to “sealing” and select the “Manual” option on the highest setting. Adjust cook time to 50 minutes.