Preheat your oven to 350°F (175°C) and grease a 9x13-inch baking dish.
Unroll one can of crescent dough and press it into the bottom of the prepared baking dish, pinching the seams together to form a solid layer.
In a large mixing bowl, use an electric mixer to beat the softened cream cheese, 1 cup of sugar, vanilla extract, and egg on medium speed until the mixture is completely smooth and creamy, about 2-3 minutes.
Spread the cream cheese filling evenly over the crescent dough layer in the baking dish.
Unroll the second can of crescent dough and carefully place it on top of the cream cheese layer, gently pressing the edges to seal.
Pour the melted butter evenly over the top layer of dough. In a small bowl, stir together the 1/2 cup of sugar and cinnamon, then sprinkle this mixture generously and evenly over the butter.
Bake for 30-35 minutes, or until the top is a deep golden brown and the center appears set.
Remove the dish from the oven and allow the cheesecake to cool completely in the pan on a wire rack before slicing and serving.