Preheat the oven to 375F, and spray a 9 inch baking dish with some nonstick cooking spray.
In a bowl, beat together the egg, egg whites, and brown sugar until it’s light and fluffy. Add in the mashed banana, yogurt, applesauce, and vanilla extract then mix well.
Stir in the all purpose flour, baking powder, and baking soda, and pour the batter into the baking dish.
Bake the cake in the oven for about 28 minutes.
Meanwhile, make the frosting by mixing together the melted butter and brown sugar. Stir in the milk then add the icing sugar and stir until it’s creamy. Store frosting in the fridge till ready to use.
Let the squares cool before removing them from the dish. Spread them with frosting and cut the cake into 16 squares, 4 down and 4 across. Enjoy!