Place a rack in the center of the oven and preheat the oven to 350 degrees F. Set aside an ungreased 13- x 9-inch baking pan.
For the crust, place the cake mix, melted butter, and egg in a large mixing bowl. Blend with an electric mixer on low speed for 2 minutes. Stop the machine and scrape down the sides of the bowl with a rubber spatula. The batter should come together in a ball. With your fingertips, pat the batter evenly over the bottom of the pan, smoothing it out with your finges. Set the pan aside.
For the filling, place the cream cheese in the same mixing bowl and with the same beaters (no need to clean either) blend with an electric mixer on low speed until fluffy, 30 seconds. Stop the maching and add the eggs, vanilla, and melted butter and beat on medium speed for 1 minute. Stop the machine and add the confectioners' sugar. Beat on medium speed until sugar is well incorporated, 1 minute. Stop the machine and scrape down the sides of the bowl with a rubber spatula.
Pour the filling onto the crust and spread with rubber spatula so that the filling covers the entire surface and reaches the sides of the pan. Place the pan in the oven.
Bake until well browned but the center still jiggles when you shake the pan, 45-47 minutes. Remove the pan from the oven and place it on a wire rack to cool, 30 minutes. Cut into squares and serve.
Cake may be stored, covered in aluminum foil, at room temperature for up to 4 days or in the refrigerator for up to 1 week. It may be frozen for up to 6 months. Thaw overnight in the refrigerator before serving.