Shred Oaxaca and Monterey Jack cheese. Chop leftover chicken, cilantro, and jalapeños. Juice lime.
In a small bowl, combine sour cream, cilantro, chopped jalapeños, pickling liquid, lime juice, and salt. Stir and refrigerate until ready to use.
Warm a skillet over medium heat. Place a tortilla in the skillet and spread a thin layer of butter on the top side. Flip tortilla so buttered side is down.
On half of the tortilla, layer shredded cheese, chopped chicken, and a sprinkle of cilantro, then top with more cheese. Fold tortilla in half.
Cook until bottom is golden and cheese begins to melt. Flip and cook the other side until golden brown and crisp. Repeat with remaining tortillas.
Slice quesadillas and serve hot with the Cilantro Jalapeño Lime Sauce on the side.