Mix apple juice, apple cider vinegar, and broth in a bowl and set aside.
Pork shoulder often comes pre-cut and tied together. If your pork is not cut, cut it in half. I find it easier to sear pork in an Instant pot as one piece tied together because of the smaller space. Just untie it after searing, before cooking.
Combine all the seasoning ingredients in a bowl and mix well. Place pork shoulder on a sheet of parchment paper and season it generously on all sides.
Preheat Instant Pot on "saute" setting and add some oil. Sear seasoned pork on all sides until browned. Use metal tongs to turn the meat. Turn saute setting off when pork is browned.
Pour apple juice mixture into the pot.
Close the lid and make sure it’s latched. Turn the valve to “seal” and set the Instant Pot to cook on high pressure cook mode for 70 minutes.
Turn off the Instant Pot when it’s done and let it naturally release for 10 minutes. You can carefully do manual release after that.
Take all the meat out into a mixing bowl and let it cool.
Pour the juices from cooking meat out of the Instant Pot into a separate container. Pour it through a fine mesh strainer.
Set the Instant Pot to "saute" and combine the ingredients for BBQ sauce in the pot. Bring it to simmer and slowly stir while it simmers for a few minutes. Turn off the Instant Pot and set sauce aside, off the heat. (Leftover juices can be used to make gravy.)
Pull pork apart once it's cooled enough to be handled. As you're working through it, discard excess fat and membrane.
Combine pulled pork meat and BBQ sauce until thoroughly mixed together.