01 - Preheat oven to 350°F (175°C). Grease and flour two 8-inch round cake pans, then line the bottoms with parchment paper.
02 - Melt 3/4 cup unsalted butter in a light-colored saucepan over medium heat. Swirl the pan frequently until the butter turns golden brown and releases a nutty, toasted 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 until evenly distributed.
04 - In a large bowl, combine the mashed bananas, granulated sugar, brown sugar, eggs, buttermilk, vanilla extract, and cooled brown butter. Stir until thoroughly blended.
05 - Gradually fold the dry ingredient mixture into the wet ingredients using a spatula, mixing just until combined. Avoid overmixing to keep the cake tender.
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 cool in the pans for 10 minutes, then invert onto wire racks to cool completely.
07 - While the cakes bake, combine 1 cup granulated sugar and 1/4 cup water in a medium saucepan over medium heat. Do not stir — allow the mixture to cook 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, expect vigorous bubbling. Stir in the flaked sea salt and let cool to room temperature.
08 - Beat 1/2 cup softened butter with an electric 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 1-2 tablespoons milk or cream, then beat until the frosting is smooth and spreadable.
09 - Place the first cake layer on a serving plate. Spread a generous layer of caramel frosting over the top and drizzle with salted caramel sauce. Position the second layer on top. Frost the top and sides of the cake evenly, then drizzle with additional caramel sauce. Finish with a sprinkle of flaked sea salt if desired.