To prepare the crumble: Dice the butter and place in a small saucepan over medium heat. Allow the butter to melt and, stirring all the while, brown the butter. You’ll remove the butter from the heat when you notice it becomes nutty in fragrance and auburn-colored flecks appear at the bottom of the pan. Don’t let the butter burn and see notes for more on this. Pour the butter into a bowl and add the brown sugar, salt, vanilla and cinnamon. Stir in the flour until large crumbles appear and then place in the fridge to chill while you prepare the cake.
To prepare the cake: Preheat the oven to 350 degrees.
In a large bowl or the bowl of a stand mixer, cream the butter, sugar, and brown sugar for one minute. Scrape the sides of the bowl and then beat in the egg until pale, about 2 minutes. Add the buttermilk, vanilla, and bananas and stir just to combine. Add the flour, baking soda, baking powder, cinnamon, and salt, and stir on low just until combined. Grease a small baking dish (I use an 8”x4” pan but you can substitute whatever you have) and spread in the butter. Use your fingers to crumble the brown butter topping evenly over top of the batter. Bake in the preheated oven until the cake has fluffed and a toothpick inserted into the center comes out clean, about 30-35 minutes depending on your pan. Allow to cool and then drizzle with the topping if desired. To prepare the topping, combine the powdered sugar, vanilla, and milk in a small bowl until smooth. Add a smidge more milk if the mixture is too thick. Drizzle over top of the cooled cake.
For the drizzle (optional):
Combine 1 cup powdered sugar, 1/2 teaspoon vanilla extract, and 2 tablespoons of milk in a large bowl and drizzle over the cooler cake.