Preheat the oven to 350°F. Spray a 9x13 baking dish with nonstick baking spray and set aside.
In a medium mixing bowl at medium speed, beat eggs with a whisk attachment for 5 minutes, until very thick and light yellow.
With the mixer still running, gradually add granulated sugar and continue whisking for five additional minutes. The mixture will be thick and light in color.
Add brown sugar and mix a few minutes longer.
In a separate bowl, combine flour, salt, and baking powder. Add the dry ingredients to the egg/sugar mixture and beat at low speed just until combined.
Heat the milk and butter in the microwave (or on the stovetop) until the butter is melted and the milk is scalding hot.
Stir the vanilla extract into the milk mixture. Pour the hot milk mixture into the batter and beat at low speed until combined.
Pour batter into the prepared baking dish.
Bake approximately 30 to 35 minutes, or until a toothpick inserted into the center of the cake comes out with just a few crumbs, and the cake has pulled away from the edges of the pan.
Cool cake in the pan and dust with powdered sugar just before serving.