Preheat the oven to 350°F (175°C). Grease a 9×13-inch baking pan or line it with parchment paper.
Mix ½ cup sugar with 2 tablespoons cinnamon until evenly combined.
Sprinkle a thin layer of the cinnamon-sugar mixture over the bottom of the prepared pan.
Unroll one can of crescent dough and press it evenly into the bottom of the pan. Pinch the seams closed.
Beat the softened cream cheese, remaining ½ cup sugar, egg, and vanilla until smooth and creamy.
Pour the cheesecake filling over the dough and spread evenly using a spatula.
Carefully place the second sheet of crescent dough over the cheesecake filling and pinch any seams together.
Brush the dough with melted butter and sprinkle the remaining cinnamon-sugar mixture evenly over the top.
Bake for 30–35 minutes, or until the top is golden brown and the center is slightly set.
Allow the bars to cool at room temperature for 1 hour.
Refrigerate for at least 2 hours, or overnight for the cleanest slices.
Cut into 16 bars using a sharp knife, wiping the blade clean between cuts. Serve chilled or slightly softened at room temperature.