Whisk together mayonnaise, minced jalapeños, jalapeño brine, and cumin in a small bowl until smooth.
Spread half the jalapeño sauce on one tortilla, then layer with half the cheese, all the diced chicken, remaining cheese, and top with the second tortilla.
Heat a large skillet over medium heat and melt half the butter.
Place the quesadilla in the skillet and cook for 3 to 4 minutes per side, adding remaining butter when flipping, until golden and cheese is fully melted.
Cut into wedges and serve with remaining jalapeño sauce for dipping.