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COOK THE PORK: In a bowl, stir together the garlic powder, onion powder, smoked paprika, salt, and black pepper. Rub it all over the outside of the pork.
Add the oil to the Instant Pot and turn to SAUTE. Once the oil is hot, add the largest of the pork pieces and brown on two sides. This will take about 2 minutes per side. If your Instant Pot issues a “hot” or “burn” warning, turn the heat down (if you have that option), or if not, turn it off entirely and sear the pork in the residual heat. If the pot cools too much to brown, turn it back on. Remove the pork to a plate. Repeat with the remaining 2 pieces—you can cook the last 2 pieces at the same time as long as they do not overlap. Remove to the plate.
Add half of the broth to the Instant Pot. With a wooden spoon, scrape up all of the stuck-on bits of food and seasoning.
Add the remaining broth and the onion.
Return the pork to the Instant Pot. Cover, seal, and cook on HIGH (manual) pressure for 40 minutes (for pork loin) or 1 hour (for pork shoulder/butt).
Once the pressure has built and the cook time has elapsed, allow the pressure to release naturally for 15 minutes (do not open right away or the meat may be tough). Vent to release any remaining pressure then carefully open the Instant Pot.
Remove the pork to a large bowl. With a slotted spoon, scooped out most of the onions and add them to the bowl with the pork. Carefully pour all but 1 cup of the pork cooking juices into an extra-large measuring cup or bowl with a spout. Keep the extra liquid handy.
PREPARE THE BBQ SAUCE: Turn the Instant Pot to SAUTE. Whisk in the tomato sauce, tomato paste, vinegar, syrup, molasses, Worcestershire, paprika, chili powder, mustard, garlic powder, black pepper, salt, and cayenne. Cover the pot loosely with the Instant Pot lid, still leaving some room for steam to escape (this sauce loves to splatter). Let simmer, whisking every few minutes, until thickened, about 5 minutes.
Once cool enough to handle, shred the pork with two forks or with your fingers. Stir the shredded pork, onions, and any juices at the bottom of the bowl into the pot with the sauce. If the sauce is too tight to easily coat the pork, splash in additional cooking juices so that the pork is nicely coated with sauce. Taste and adjust seasoning as desired. Pile onto buns, top with slaw, and enjoy!