Replicating the item in a home kitchen requires nailing the emulsified sauce before assembling the ingredients. Store-bought cheddar alone fails because it releases too much oil under high heat, softening the tortilla.
The Sauce Formulation:
Whisk together 1/2 cup mayonnaise, 1 tablespoon sour cream, 2 tablespoons finely minced pickled jalapeños, 1 teaspoon jalapeño brine, 1/2 teaspoon ground cumin, 1/2 teaspoon garlic powder, 1/2 teaspoon onion powder, 1/2 teaspoon smoked paprika, and 1/4 teaspoon salt. Refrigerate for at least two hours to allow the spices to hydrate.
The Assembly:
Brush a 12-inch flour tortilla lightly with vegetable oil on one side. Lay it dry-side down in a heavy cast-iron skillet over medium-low heat. Spread two tablespoons of the prepared sauce over one half. Layer 1/3 cup shredded Monterey Jack, 1/3 cup shredded sharp cheddar, and 3 ounces of grilled, diced chicken breast seasoned with chili powder and lime juice. Fold the dry half over the fillings.
Weigh the quesadilla down with a second heavy pan. Toast for roughly 3 to 4 minutes per side until deep golden-brown ridges form. Remove, rest for two minutes on a cutting board to set the cheese, and slice into four distinct wedges using a rocking pizza cutter.