In a small bowl, combine the smoked paprika, brown sugar, garlic powder, onion powder, chili powder, salt, and black pepper. Pat the pork shoulder chunks dry with paper towels, then rub the spice mixture generously all over each piece.
Set the Instant Pot to saute mode and add the olive oil. Once hot, sear the pork in batches until browned on all sides, about 3 minutes per side. Remove and set aside.
Add the sliced onion to the pot and cook for 3 minutes, stirring occasionally. Add the garlic and cook for 30 seconds until fragrant. Pour in the apple cider vinegar and chicken broth, scraping the bottom of the pot to deglaze any browned bits.
Return the pork to the pot and pour 1/2 cup of the BBQ sauce over the top. Lock the lid in place and set the valve to sealing. Pressure cook on high for 60 minutes.
When the cook time is complete, allow a natural pressure release for 15 minutes, then carefully switch the valve to venting to release any remaining pressure. Remove the lid once the pin drops.
Transfer the pork to a large bowl or cutting board. Using two forks, shred the meat, discarding any large pieces of fat.
Drain most of the liquid from the pot, leaving about 1/2 cup behind. Return the shredded pork to the pot and add the remaining 1 cup of BBQ sauce. Stir everything together and set to saute mode for 3 to 4 minutes, letting the sauce thicken and coat the pork.
Serve the pulled pork piled high on hamburger buns with extra BBQ sauce on the side if desired.