Combine mayonnaise, sour cream, pickled jalapeno juice and minced jalapenos in a bowl along with garlic, cumin, smoked paprika, cayenne, salt, and pepper; stir well until smooth and taste to adjust seasoning.
Shred the rotisserie chicken if using and mix it into a medium bowl with some or all of the jalapeno sauce until the chicken is nicely coated and creamy; reserve extra sauce for dipping.
Heat a skillet to medium, place one flour tortilla in the pan, sprinkle a layer of cheese, add some chicken mixture, top with more cheese, and cover with a second tortilla.
Cook the quesadilla until the bottom tortilla turns golden brown, about 1 minute, then carefully flip and cook the other side for 30 to 60 seconds until golden and cheese melts.
Repeat the process with the remaining tortillas and filling, slice each quesadilla into wedges as desired, and serve immediately while warm.
