Begin by preparing the jalapeño sauce in a mixing bowl by combining mayonnaise, sour cream, pickled jalapeño juice, minced jalapeños, minced garlic, cumin, paprika, cayenne pepper, salt, and pepper. Stir well until thoroughly blended.

Shred rotisserie chicken into bite-sized pieces and mix half of the jalapeño sauce with the chicken in a large bowl. Set aside remaining sauce for dipping.

Heat a skillet over medium heat, place one tortilla and sprinkle cheese on one half. Spoon chicken mixture over cheese, fold, and cook until golden brown, about 2-3 minutes.

Flip quesadilla carefully and cook until golden brown on the other side for another 1-2 minutes. Repeat with remaining tortillas and filling.

Slice quesadillas into wedges and serve alongside reserved jalapeño sauce for dipping.