Heat a 7¼-quart Dutch oven over medium heat, add the oil and heat through. Add the onion and sauté, stirring occasionally, until the onion is tender and opaque, about 5 minutes. Remove from the pan and set aside.
Season the pork with salt and pepper. Working in batches, add the pork and sear until golden brown on all sides, about 5 minutes. In a small bowl stir together half of the gochujang and soy sauce.
Add the onions back to the pan with garlic, gochujang mixture and the beef stock. Raise the heat to high and bring to a boil. Cover and move to the oven. Cook, stirring every 30 minutes, until the pork is fork tender, about 2 hours.
Use a spoon to skim the oil from the top and discard. Remove the meat from the broth to a large mixing bowl and gently shred the meat, discarding and fatty pieces. Stir in the remaining gochujang and lime juice.
For assembling the quesadilla:
Heat a large non-stick skillet over medium heat. Add one tablespoon of the butter and allow to melt. Add one of the tortillas and evenly top with ½ cup of Monterey Jack cheese, ½ cup of the Cheddar, 1 cup of the shredded meat with sauce and ¼ cup of the kimchi. Top with the other tortilla and cook, flipping once, until the quesadilla is golden brown on both sides, about 3 minutes per side.
Remove from the pan and slice into 6 equal slices. Repeat with the remaining ingredients.