Preheat the oven to 350°F and line a 9-inch square baking pan with parchment paper. Whisk the flour, granulated sugar, baking powder, and salt together in a bowl.
Stir the melted butter, whole milk, and vanilla extract into the dry ingredients until a soft, thick dough forms. Press the dough evenly across the bottom of the prepared pan.
Beat the cream cheese until completely smooth. Beat in the powdered sugar, melted butter, sour cream, cornstarch, vanilla extract, and salt until the filling is creamy and lump-free.
Spread the cream cheese mixture evenly over the cake base, taking care not to mix the two layers together.
Bake for 32 to 38 minutes, until the edges are golden and set but the center still has a gentle wobble. Do not bake until the center is completely firm or the signature gooey texture will be lost.
Cool completely in the pan for at least 2 hours. Dust with the remaining powdered sugar before serving.
