In a small bowl, mix the cheeses together. Top each of the 6 tortillas with 1/4 cup of the cheese mixture, 1/4 cup of the chicken, and another 1/4 cup of the cheese mixture. Sprinkle with the cilantro and top with the remaining 6 tortillas. Press down gently on the quesadillas to compress them.
Heat a large griddle or skillet over medium-low heat. Add 3 tablespoons of the oil, then add 3 of the quesadillas and cook for about 3 minutes per side, or until they are golden brown and crisp on the outside and the cheese has melted. Transfer the quesadillas to plates and sprinkle with salt. Wipe out the pan and cook the remaining 3 quesadillas, using the remaining 3 tablespoons oil.