Thinly slice the apples into 24 1/4-inch-thick slices.
Sprinkle 2 tablespoons cheese over one half of the tortilla. Place four apple slices, barely overlapping, on top of cheese. Then sprinkle 2 more tablespoons of cheese on top of the apples and fold the tortilla in half. Repeat with remaining tortillas.
Heat 1 tablespoon butter in a large skillet over medium-high heat and cook the quesadillas until the cheese melts and the tortillas are golden brown on both sides, about 2 minutes per side. Repeat with the remaining butter and tortillas.