In a 7-quart dutch oven, add the first 11 ingredients. Fill with water almost all the way to the top and bring to a boil. Boil for 30 minutes, removing any impurities on the top using a spoon or ladle. (It will look like a light brown foam around the edges of the pot)

Reduce heat to medium-low and bring to a simmer. Remove the peppers and transfer them to a blender or food processor along with 1 cup of the cooking liquid. (The peppers should have softened by now) Blend until smooth and add back into the pot and stir.

Cover and simmer for 3 hours, stirring once every hour. The meat will be done when it is easily shreddable on its own. Remove from the heat and let sit for 5 minutes.

Using a ladle or spoon, carefully skim off the fat floating on the top and transfer it to a small bowl. ( this is what the corn tortillas will be cooked in)

Using tongs, transfer the meat to a large bowl and remove the bones and shred the meat. Set aside.

Using a large bowl and strainer, strain out the dutch oven into the bowl and discard anything left in the stainer. Save the cooking liquid for dipping the tacos.

Heat a cast iron skillet over medium heat. Dip a corn tortilla into the reserved fat and place it on the skillet. Top with 3 tbsp of the quesadilla cheese and about 4 oz of the shredded meat, fold it over and cook on both sides for 2 minutes or until crispy. Transfer to a paper towel-lined plate and repeat with the remaining tortilla shells.

Serve with a cup of the cooking liquid and top with chopped onion, cilantro and limes if desired.