Heat 1 teaspoon vegetable oil in a large cast iron skillet over medium low heat. Working one at a time, add 1 tortilla to the skillet and sprinkle with 1/3 cup cheese mixture in an even layer.
Top with 1/3 cup chicken on half of the tortilla; season with salt and pepper, to taste.
Cook until the cheeses have melted and the underside is golden brown, about 3-4 minutes; gently fold in half, pressing lightly. Keep warm in oven up to 20 minutes. Repeat with remaining vegetable oil and tortillas to make 6 quesadillas.