Preheat oven to 350 F. (regular bake setting/not fan assisted) and grease an 8x4-inch loaf pan. Set aside. Melt butter and set aside to cool slightly while you proceed with the recipe.
In a large bowl, whisk together the flour, white sugar, brown sugar, baking soda and salt well, breaking up any lumps of brown sugar. In a separate medium bowl, lightly beat the eggs with a fork. Add the mashed bananas, vanilla and slightly cooled melted butter and stir just until well combined. Add the banana mixture to the bowl with the flour mixture and stir with a spoon just until combined. Spoon in to prepared pan.
Bake in preheated oven for about 60 minutes, or until a tester inserted in the middle comes out clean. Allow to cool in the pan for 15-20 minutes, then run a knife around the outside and remove loaf to a cooling rack. Allow to cool completely before frosting.
Prepare the chocolate buttercream: In a medium bowl with an electric mixer or the bowl of a stand mixer with the paddle attachment, cream the butter well. Turn the mixer off. Add the milk, cocoa, icing sugar and a pinch of salt. Turn the mixer on low until the mixture is moistened, then increase speed to medium and mix until creamy, adding more icing sugar if needed to make a stiff, but still creamy frosting. Spoon on to completely cooled banana bread and use the end of an offset spatula to make ridges by running lines down the top from one end to the other.
Notes
Be sure to read the notes above this Recipe Card, for more tips on making this recipe!