Whisk together mayonnaise, minced jalapeños, jalapeño brine, and cumin until smooth, then set aside.
Spread half the cheese on one half of each tortilla, top with diced chicken, then fold the tortilla over to form a half-moon.
Heat vegetable oil in a large skillet over medium heat until shimmering.
Cook each quesadilla for 3 to 4 minutes per side, pressing gently with a spatula, until golden and the cheese is fully melted.
Cut each quesadilla into 3 wedges and serve with the creamy jalapeño sauce for dipping or drizzling.