Chop the pork shoulder into small pieces approximately 3-inches each. Place into a bowl and set aside.
In a large bowl, whisk together the apple cider vinegar, ½ cup of the BBQ sauce, and ½ cup of the beef broth. Set aside.
In a large bowl, whisk together the paprika, garlic powder, onion powder, kosher salt, and pepper until combined. Season the pork.
Turn your pressure cooker to SAUTE and HI. Heat the olive oil in the pressure cooker pot for just about 2 minutes, until it heats up. Once the pot is hot, add the seasoned pork and saute for 4 to 6 minutes, stirring constantly with a wooden spoon or large spoon.
Once the meat has been browned on all sides, determine if the pot needs to be de-glazed: if there are brown bits on the bottom of the pot under the meat, remove the meat and transfer to a clean large bowl. Then pour a tablespoon or two of the beef broth into the pot, and use a wooden spoon to scrape up all the brown bits at the bottom.If, however, there are no brown bits at the bottom of the pot, pour in the chicken broth and the BBQ sauce mix. Add the meat and stir everything together until the meat is well coated.
Seal the pressure cooker lid, making sure the pressure release valve is in the SEAL position, until the lid is locked closed. Select PRESSURE/MANUAL and HI, then set the time for 40 minutes. The pressure cooker will take about 10 to 15 minutes to come to pressure, and then start the 40 minutes of cooking.
Once the pressure cooking is complete, natural release* for 10 minutes and then quick release. Once all pressure has been released and the pressure release valve is all the way down, carefully open the lid.
You may have excess liquid in the pot. If you do, carefully strain out the meat, draining the excess liquid into a sick, then transfer the meat to a new large bowl or return the meat to the pot. Using two forks, shred the pork. Pour in the remaining BBQ sauce and toss together until the pork is well coated.
Notes
Resting: Let the meat rest for at least 15 minutes after cooking before you pull it apart to let the juices sink in and so the meat doesn't dry out.