Remove pork shoulder from refrigerator at least 30 minutes, but not more than 2 hours, before cooking. Set aside.
Prepare dry rub by combining all ingredients in a small bowl. Stir to combine and rub the dry rub into all sides of each pork shoulder section, reserving one tablespoon of the rub for later use. Set the seasoned pork and reserved dry rub aside.
Select the “Sauté” function on the Instant Pot® and add one tablespoon olive oil once hot. Add seasoned pork and sear on each side, approximately 2-3 minutes per side. (Work in two batches, if necessary). Transfer pork to a plate and set aside.
Add remaining olive oil, along with celery, carrots, and yellow onion. Sauté, stirring occasionally, until vegetables start to develop some color and become fragrant, approximately 4-5 minutes.
Add broth and stir to combine, gently scrapping any brown bits off the bottom of the pot with a wooden spoon in the process. Turn the unit off.
Return pork shoulder to the Instant Pot®. Add bay leaves, rosemary, and thyme. Sprinkle reserved tablespoon of dry rub on top and stir to combine. Add lid and lock into place. Switch the vent to “Sealing” and set the “Manual” setting to 50 minutes.
When cook time is complete, allow pressure to release naturally for 20 minutes, then manually release any remaining pressure, if necessary.
Remove lid and discard the bay leaves and woody stems from the rosemary and thyme. Transfer the pork roast to a serving platter and cover to keep warm. Set aside.
With an immersion blender, blend the pan sauces until smooth. Taste and adjust seasonings, as desired. Serve pork roast immediately with choice of sides and topped with some of the warm sauce. Enjoy!