Preheat oven to 350 degrees F (180 degrees C). Line a 8 inch round cake pan and grease the sides.
Beat Unsalted Butter (125 gram) and Brown Sugar (120 gram) till light and fluffy. Beat in Egg (2) one at a time. Stir in mashed Banana (250 gram) and Vanilla Extract (1 teaspoon) till combined.
Mix Baking Soda (1 teaspoon), Baking Powder (1 teaspoon), and hot Milk (2 tablespoon) together before stirring into the banana mixture. Sift in Cake Flour (230 gram) and baking soda, fold well with a rubber spatula.
Pour banana batter into prepared cake pan and bake in the preheated oven for about 40-45 minutes or a skewer inserted comes out clean.
Set aside to cool cake in pan for 5-10 minutes before lifting out onto rack to cool.
For the Chocolate icing, melt the Unsalted Butter (30 gram) in a small pot at low heat. Pour in Unsweetened Cocoa Powder (3/2 tablespoon) and mix well. Then add the Powdered Confectioners Sugar (6 tablespoon) stir well and turn off the heat.
Add Vanilla Extract (1/4 teaspoon), and Milk (1 1/2 tablespoon) little by little into the pot. Stir mixture until it is smooth and it has spreading consistency you desire.
Spread chocolate icing on top of cake and chill the cake for about 30 minutes.