Position an oven rack in the center of the oven and preheat to 350 degrees F.
Unroll one can of dough and lay it in the bottom of a 9-by-13-inch glass baking dish. Use your fingers to press the dough into the bottom of the dish as evenly as you can, pinching the perforations to seal as needed.
Beat together the cream cheese, vanilla, egg and 1 cup of the sugar in a medium bowl with an electric hand mixer on medium speed until smooth and fluffy, 1 to 2 minutes. Use a rubber spatula to spread the cream cheese mixture over the dough. Unroll the second can of dough and lay it on top of the cream cheese layer. Stretch the dough as needed to cover the cream cheese and use your fingers to pinch the perforations to seal. Do not worry if the dough is not perfectly even; that will happen as it bakes. Pour the butter on top of the dough and use a spoon to spread it out evenly. Stir together the remaining 1/3 cup sugar and the cinnamon in a small bowl and sprinkle all over the top of the dough.
Bake until the dough is baked through and the cinnamon sugar has formed a crust on top of the dough, about 30 minutes. Cool for 15 minutes before cutting into 12 pieces. The bars can also be cooled completely before cutting and serving.
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