Season a whole chicken with salt and pepper on the inside and out. Set it in an Instant Pot or other pressure cooker with 2 cups of water, and cook for 1 hour. After the chicken has been thoroughly cooked, let it cool down naturally in the pressure cooker.
Once the chicken has cooled enough to handle, take it out and put it on a wood cutting board. Pick the meat off the bones and dice it up. (Save the bones and cooking liquid for making chicken bone brother later!) Taste the meat for seasoning.
In a cast iron skillet on top of the stove, add olive oil and heat to medium-low or medium heat. While it's heating, get your flour tortillas and cheese ready.
Once the skillet is hot, place a tortilla in the skillet, spread chicken on top, and add additional salt and pepper, if desired. Then add slices of cheese to cover the chicken, and add a second tortilla on top. Let it cook for about 3 minutes until the tortilla is golden brown. Then flip it over and cook an additional 3 minutes.
When the cheese is melted and the tortillas are golden brown, remove the quesadilla to a plate and start another quesadilla by following the same steps.
Let the quesadillas cool for a couple minutes, and then cut into wedges using sharp kitchen scissors. Serve with sliced avocado, salsa, sour cream, or fresh cilantro, as desired. It serves 4 people as a main course, or 6-8 people as a side dish.