1. In a small bowl, whisk together the mayonnaise, minced pickled jalapeños, jalapeño juice, sugar, cumin, paprika, cayenne pepper, and garlic powder. Season with salt and pepper to taste. Set the sauce aside.

2. In a separate bowl, combine the cooked, diced or shredded chicken with half of the prepared sauce, stirring until the chicken is evenly coated.

3. Heat a skillet or griddle over medium heat. Lightly grease the surface with a small amount of cooking oil or spray.

4. Place one tortilla in the skillet and on one half, layer about 1/2 cup of shredded cheese, followed by 1/2 cup of the chicken mixture, and then top with another 1/2 cup of cheese. Fold the other half of the tortilla over the filling to create a half-moon shape.

5. Cook the quesadilla for 2-3 minutes per side, or until the bottom is golden brown and crispy. Carefully flip the quesadilla using a spatula and cook for another 2 minutes, or until the cheese is melted and the tortilla is crisp.

6. Repeat the assembly and cooking process with the remaining tortillas, chicken, and cheese to make the rest of the quesadillas.

7. Once all the quesadillas are cooked, cut each one into wedges and serve with the remaining sauce on the side for dipping.