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In a small bowl, stir together all spice rub ingredients until well combined.
Cut the pork into 4 pieces and trim off excess fat. Rub the pork on all sides with the spice rub.
Pour olive oil into Instant Pot insert and turn it on saute mode. When the pot is hot, place the pork in the Instant Pot and brown on all sides. You may need to do this in two batches. Transfer the browned pork to a clean plate.
Turn off Instant Pot. Pour about half of the chicken broth into the pot and use a rubber spatula to scrape up any browned bits off of the bottom of the pot. Add the rest of the broth, the barbecue sauce, apple cider vinegar and honey (or brown sugar). Stir to combine.
Return the pork to the Instant Pot. Close the lid. Turn the steam release valve to the sealing position. Set the Instant Pot to pressure cook at high pressure for 60 minutes.
The Instant Pot will take about 15 minutes to pressurize and then the cook time will begin. When the cook time is done, allow the pressure to release naturally for 20 minutes (just leave the Instant Pot alone). Then, quick release any remaining pressure by carefully turning the steam release valve to the venting position (use either the handle of a long spoon or a potholder to turn the valve).
Transfer pork to a large bowl.
In a small bowl, stir together cornstarch and water until dissolved. Stir the cornstarch mixture into the sauce in the Instant Pot. Turn the Instant Pot on saute and cook the sauce, stirring occasionally, until slightly thickened, 4-5 minutes. Turn off the Instant Pot.
Meanwhile, use two forks to shred the pork. Stir it back into the thickened sauce. Serve pork on burger buns topped with additional barbecue sauce, if desired.