Combine the cheeses in a medium bowl.

Heat a large frying pan over medium heat until hot, about 3 minutes. Place a tortilla in the pan and sprinkle with half of the chicken, half of the cilantro, and and half of the cheese mixture. Top with a second tortilla and cook until the underside of the bottom tortilla is golden brown in several spots and half of the cheese is melted, about 3 minutes. Using a spatula, carefully flip the quesadilla over and cook until the underside of the second tortilla is crisp and golden brown in several spots and all of the cheese is melted, about 2 to 3 minutes more.

Slide the quesadilla from the pan onto a cutting board and cut into wedges. Repeat with the remaining ingredients to make a second quesadilla. Serve topped with guacamole, salsa, and sour cream, if desired.