Rub Pulled Pork Dry Rub: Mix all the dry rub ingredients and rub it all over the pork shoulder picnic pieces. Then, put the pork shoulder picnic in the fridge for 30 minutes to overnight.
Heat up Pressure Cooker: Heat up your pressure cooker (Instant Pot: press Sauté button). Make sure your pot is as hot as it can be when you place the pork shoulder meat into the pot (Instant Pot: wait until the indicator says HOT).
Optional Step - Brown Pork Shoulder (See Tips): Add 1 tbsp (15 ml) olive oil into the pot. Ensure to coat the oil over the whole bottom of the pot. Add the pork shoulder pieces into the pot. Brown the pork shoulder on all sides. Remove and set aside.
Add BBQ Sauce and Deglaze: Pour in half of the Pulled Pork BBQ Sauce and deglaze the bottom of the pot. Then, add in the remaining BBQ sauce mixture.
Pressure Cook Pork Shoulder (See Tips): Place all the pork shoulder pieces into the pot. If you left the skin on, make sure the skin side is facing up. Close lid and pressure cook at High Pressure for 60 minutes (see notes). Turn off the heat and fully Natural Release (roughly 15 minutes).
Fork Tender Check: Open the lid carefully. Take one piece of pork shoulder out and see if you can shred through the meat easily with two forks. If it is not fork-tender, cook for an additional 10 to 20 minutes at High Pressure, fully Natural Release.
Shred Pork & De-Fat the Sauce: Use some cool Pulled Pork Shredder Claws or two regular forks to shred the pork shoulder meat. Use a fat separator to separate the fat from the BBQ sauce.
Season, Reduce, Serve: Reduce the BBQ sauce to your desired thickness. Taste the BBQ sauce and add in additional brown sugar or kosher salt if desired. Place the pulled pork back into the BBQ sauce. Mix well and serve!