01 - Preheat oven to 350°F (175°C). Grease and flour two 8-inch (20-cm) round cake pans. Line bottoms with parchment paper for easy release.
02 - In a medium saucepan, melt the butter over medium heat. Continue cooking, swirling the pan frequently, until the butter turns golden brown and gives off a nutty aroma, about 5–7 minutes. Immediately pour into a heatproof bowl and let cool slightly.
03 - In a medium bowl, whisk together the all-purpose flour, baking soda, and salt. Set aside.
04 - In a large bowl, combine the mashed bananas, granulated sugar, brown sugar, eggs, buttermilk, vanilla extract, and the browned butter. Stir until well blended and smooth.
05 - Gradually fold the dry ingredient mixture into the wet ingredients using a spatula, mixing until just combined. Take care not to overmix, as this will make the cake dense.
06 - Divide the batter evenly between the two prepared cake pans. Bake for 35–40 minutes, or until a toothpick inserted into the center comes out clean. Let the cakes cool in the pans for 10 minutes, then turn out onto wire racks to cool completely.
07 - While the cakes bake, combine the sugar and water in a medium saucepan over medium heat. Do not stir—allow the mixture to cook undisturbed until it turns a deep amber color. Remove from heat and carefully stir in the cubed butter until fully melted. Slowly whisk in the room-temperature heavy cream, watching for splatter. Stir in the flaked sea salt. Let the sauce cool to room temperature.
08 - In a mixing bowl, beat the softened butter with a hand mixer or stand mixer until light and creamy. Gradually add the sifted powdered sugar and a pinch of salt, beating on low until incorporated. Add 4–5 tablespoons of the prepared salted caramel sauce and beat until smooth. Adjust consistency with milk or cream, 1 tablespoon at a time, until the frosting is spreadable.
09 - Place one cake layer on a serving plate. Spread a generous layer of caramel frosting over the top and drizzle with salted caramel sauce. Place the second cake layer on top. Spread the remaining frosting over the top and sides of the cake. Drizzle with additional caramel sauce and garnish with flaked sea salt if desired.