Melt 448g of unsalted butter in a saucepan over medium heat for 10 to 12 minutes until foamy and golden brown. Set aside 140g for the cake and the rest for the frosting, and chill both portions until softened.
Preheat your oven to 175 °C. Grease and line a 23cm x 23cm baking tin.
For the batter, whisk flour, bicarbonate of soda and salt together. In a separate bowl, beat 140g brown butter until pale and fluffy. Add the sugar and beat for two minutes.
Add the eggs and vanilla and mix until smooth. Stir in the mashed bananas and buttermilk, then mix in the dry ingredients until just combined.
Pour into the prepared tin and bake for 40 to 45 minutes. Once done, leave the cake to cool in the tin for a little bit, then transfer to a wire rack and leave to cool completely.
For the frosting, beat the remaining brown butter for five to 10 minutes until pale and fluffy. Gently mix in the cream cheese and icing sugar until smooth and fluffy.
Spread the frosting over the cooled cake and drizzle caramel sauce. Swirl the caramel through the frosting, then cut into squares and serve.