In a small bowl, combine 1 tablespoon cinnamon with 1/2 cup sugar and mix well. Set aside.
Preheat oven to 350 degrees F and position rack in the middle. Grease a 9x13-inch baking pan and sprinkle 1-2 tablespoons of the cinnamon sugar mixture evenly on the bottom.
Open one tube of crescent roll dough and unroll carefully. Press seams together to form a single sheet and place in the prepared pan, covering the bottom completely.
In a mixing bowl, beat the softened cream cheese, 3/4 cup sugar, egg, and vanilla extract until smooth and creamy, about 2 minutes. Do not overmix.
Pour the cheesecake filling over the base layer and spread evenly with a spatula.
Open the second tube of crescent dough, seal seams, and gently place over the cheesecake filling to form a top layer.
Brush the melted butter evenly over the top layer using a spoon or pastry brush.
Sprinkle the remaining cinnamon sugar mixture generously over the buttered top layer.
Bake for 26-30 minutes until the crescent dough is puffed and golden brown. Remove from oven and cool on a wire rack.
Once cooled to room temperature, refrigerate for at least 3 hours to allow layers to set properly. Slice into bars before serving.