Select saute on the Instant Pot and add the vegetable oil. Once heated, brown each piece of pork on both sides for around 3 minutes. Set aside.
Add 1 cup of BBQ sauce and 1/2 cup of chicken broth to the pot and mix well, making sure you get all the bits off the bottom. Add the pork and the juices. Lock the lid and set high pressure and cook for 1 hour 15 minutes.
When the cooking time ends, turn off the Instant Pot and allow the pressure to release naturally, around 20 minutes.
In the meantime, preheat the oven to 350 degrees Fahrenheit, place the French fries on the baking tray, and cook according to packet instructions.
Once done, remove from the oven and set aside.
When the valve drops on the Instant Pot, remove the pork and put it into a large bowl. Using two forks, shred the pork.
Strain the cooking liquid, reserving 1/2 a cup. Put the Instant Pot back on saute and add the shredded pork back to the pot with another cup of BBQ sauce and the reserved cooking liquid. Bring to a simmer and keep stirring for 5 minutes.
In a baking tray, baking dish, or skillet, layer half of the French fries, half of the pork, half of the red onion, half of the scallions, half of the cheddar cheese, and half of the Monterey Jack, and then repeat again with another layer.
Bake in the oven for 10 to 15 minutes until the cheese has melted. Add the sour cream and serve.