Preheat the oven to 350°F (175°C). Line a 9x13-inch baking pan with parchment paper or lightly coat with nonstick spray.
To prepare the bottom layer, stir together the yellow cake mix, one egg, and melted butter in a medium-sized mixing bowl until a thick dough-like consistency forms. Press this mixture firmly and evenly into the bottom of the baking dish.
In a separate large bowl, whip the softened cream cheese until completely smooth. Incorporate the eggs and vanilla extract, blending thoroughly. Gradually add the melted butter, then beat in the powdered sugar until the mixture is smooth and luxuriously thick.
Spread the filling evenly over the pressed crust layer, smoothing the surface with a spatula.
Bake in the preheated oven for 40 to 45 minutes. The outer edges should be set and lightly golden, while the center should remain slightly soft and gooey. Avoid overbaking to maintain the signature texture.
Allow the dessert to cool entirely in the pan. Once cooled, slice into bars or squares. A dusting of powdered sugar can be added before serving, if desired.