Preheat oven to 325 F degrees. Grease a 8 inch x 8 inch baking dish.
Start by heating the butter in a small saucepan over medium heat until melted. Whisk continuously until it becomes golden brown and tiny dark brown flecks appear in the bottom of the pan. Remove from heat, whisk for another 30 seconds and strain through a fine sieve into a bowl. Let cool to room temperature.
Separate eggs and add the egg whites to a mixer and mix until egg whites are stiff. Place egg whites in a bowl and set aside.
Beat the egg yolks with the brown sugar until light golden color. Add butter and vanilla extract and continue beating for another minute or two after which you can add the flour and mix it in until fully incorporated.
Slowly start adding the milk and beat until everything is well mixed together. The mixture will be very watery.
Add the egg whites, a third at a time and gently fold them in using a spatula, repeat until all egg whites are folded in. Another variation to folding in the egg whites would be to add a third of the egg whites and gently whisk them in to the cake batter, then reverse the process and add a bit of the cake mixture to the egg whites and gently whisk in, repeat until all cake batter has been whisked in.
Pour batter into baking dish and bake for 40 to 70 minutes or until the top is lightly golden. The baking time could vary greatly depending on the oven, so take a peek at around 40 minutes and see how it looks.
Sprinkle some powdered sugar after cake has cooled.