In a small bowl, combine mayonnaise, sour cream, pickled jalapeño juice, minced jalapeños, garlic, cumin, smoked paprika, cayenne, salt, and pepper; mix thoroughly.
Shred the rotisserie chicken and mix it with part or all of the jalapeño sauce in a medium bowl until creamy; reserve any remaining sauce for dipping.
Heat a skillet over medium heat. Place one tortilla in the skillet, top with cheese, then chicken mixture, more cheese, and finish with a second tortilla.
Cook until the bottom tortilla is golden brown, about 1 minute, then flip and cook the other side until golden, about 30 to 60 seconds.
Repeat with the remaining tortillas, then slice each quesadilla into wedges and serve immediately.