Preheat the oven to 350 F and grease a 9x13-inch baking pan. In a large bowl, whisk together the gluten free 1-to-1 baking flour, granulated sugar, baking powder, and salt. Add the melted butter, egg, and vanilla extract and stir until a thick dough forms. Press the dough evenly into the bottom of the prepared pan.
For the gooey layer, beat the cream cheese with an electric mixer on medium speed until smooth and creamy. Add the eggs one at a time, beating well after each. Mix in the vanilla extract and melted butter.
Add the powdered sugar one cup at a time, beating on low speed until fully incorporated. The mixture should be smooth and pourable.
Pour the gooey layer evenly over the cake base. Bake for 40 to 45 minutes until the top is light golden and the edges are set, but the center still has a slight jiggle. The center should look slightly underdone as it will set as it cools.
Let cool completely in the pan. The center will sink slightly and become irresistibly gooey. Dust generously with extra powdered sugar, slice into squares, and serve.