Preheat the oven to 350°F (175°C). Mix the sugar and cinnamon in a small bowl and set aside. Grease a 9x13-inch baking dish.
Roll out one can of crescent dough and press it into the bottom of the baking dish, pinching the seams together to create a solid layer. Sprinkle half of the cinnamon-sugar mixture over the dough.
In a mixing bowl, beat cream cheese, sugar, vanilla extract, and egg until smooth and creamy. Spread the cheesecake mixture evenly over the bottom layer of dough.
Unroll the second can of crescent dough and carefully place it over the cheesecake filling. Brush the melted butter over the top and sprinkle the remaining cinnamon sugar evenly.
Bake for 30 minutes or until golden brown. Let it cool for at least 30 minutes before slicing into squares. For best results, chill in the refrigerator for an hour before serving.