Heat a skillet under medium to low heat until hot. Butter one side of a large quesadilla with 1 Tablespoons of softened butter. Place the buttered side of the quesadilla on the hot skillet. Shingle the thinly sliced apples on one side of the quesadilla. sprinkle the grated cheese,walnuts,and brown sugar over the apples evenly. Fold the other half of the quesadilla over the ingredients. Cook until both sides of the quesadilla are browned, flipping the quesadilla as needed until cheese has melted and you have a golden brown color on both sides of the quesadilla. Remove from the pan, let cool to room temperature. Cut into slices, serve.