Banana Cake: In a small saucepan, brown the butter over low heat, stirring constantly. Once it begins to brown and foam, wait about 30 seconds and turn off the heat. Let it cool completely.
Heat the oven to 180 ° C / 350 °F Line the bottom of a 9-inch round pan with parchment paper, butter, and flour the pan.
In a medium bowl, add the flour, baking powder, and salt.
In a small bowl mix the sugar, brown butter, and the eggs. Mix well and add milk and mashed bananas. Add this mixture to the dry ingredients and mix.
Divide the batter evenly in the already prepared pans. Bake between 25 to 30 minutes or until the center of the cakes, when you insert the tip of a knife, comes out clean. Let them cool completely before frosting.
Dulce de Leche Filling: In a small saucepan, over medium heat, heat dulce de leche and milk until well incorporate it. Let it cool.
Dulce de Leche Frosting: In a medium-size mixing bowl, using an electric mixer, beat cream cheese until creamy. Add dulce de leche and icing sugar and beat until creamy.
Assemble the Cake: Set one cake on a cake platter. Add half the dulce de leche frosting to a ziplock bag, cut the tip and make a loop around the cake. Pour the dulce de leche filling over the cake and spread it.
Gently set the second cake on top of the first cake. Add the rest of the frosting and spread it with the help of a knife or spatula. If desired, microwave 2 tablespoons of dulce de leche for 20 seconds, and add little puddles over the frosting. Use a knife or spatula and swirl the dulce de leche, mixing it with the rest of the frosting.