Make brown butter: In a heavy-bottomed saucepan, melt two sticks of butter on medium heat, stirring occasionally. Butter will steam and foam. Watch closely, stirring as the butter turns brown. Remove to a cool bowl. Set aside.
Preheat the oven to 350 F. Grease a 9 x 13-inch cake pan. Set aside.
Make the cake batter: In a medium mixing bowl, combine cake mix, eggs, and melted butter into a dough-like batter. Press evenly into the bottom of the pan and set aside. Set aside.
Make the filling: Whisk together wet ingredients and the reserved egg. Add salt and sugar, whisking to combine. Then, add flour in increments and finish whisking until thoroughly combined. Spread the filling evenly over the crust in the pan, smoothing to the edges.
Bake for approximately 30 to 35 minutes, until the top is golden brown but the center still wobbles.
Cool completely and cut into squares. Dust with confectioners' sugar and serve.