01 - Preheat the oven to 350°F. Grease and flour two 8-inch round cake pans, then line the bottoms with parchment paper.
02 - Melt the butter in a light-colored saucepan over medium heat, swirling the pan frequently. Continue cooking until the butter turns a deep golden brown color and releases a nutty, toasted aroma, approximately 5 to 7 minutes. Immediately transfer to a heatproof bowl and allow to cool slightly.
03 - In a medium bowl, whisk together the all-purpose flour, baking soda, and salt until evenly distributed.
04 - In a large mixing bowl, combine the mashed bananas, granulated sugar, brown sugar, eggs, buttermilk, vanilla extract, and the slightly cooled brown butter. Stir until thoroughly incorporated.
05 - Gradually fold the flour mixture into the wet ingredients using a spatula, mixing only until no dry streaks remain. Avoid overmixing to keep the crumb tender.
06 - Divide the batter evenly between the two prepared cake pans, smoothing the tops with a spatula. Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean.
07 - Let the cakes rest in their pans for 10 minutes, then turn them out onto wire racks to cool completely before assembling.
08 - While the cakes bake, combine the sugar and water in a medium saucepan over medium heat. Do not stir; allow the mixture to simmer until it turns a deep amber color. Remove from heat and carefully add the cubed butter, stirring until fully melted. Slowly whisk in the room-temperature heavy cream, taking caution as the mixture will bubble vigorously. Stir in the flaked sea salt and let the sauce cool to room temperature.
09 - Beat the softened butter in a mixing bowl until light and creamy. Gradually add the sifted powdered sugar and a pinch of salt, beating on low speed until combined. Increase the speed and blend in 4 to 5 tablespoons of the prepared caramel sauce. Add milk or cream 1 tablespoon at a time until the frosting reaches a smooth, spreadable consistency.
10 - Place the first cake layer on a serving plate. Spread an even layer of caramel frosting over the top and drizzle with salted caramel sauce. Position the second layer on top, then frost the top and sides of the entire cake. Finish with additional caramel drizzle and a sprinkle of flaked sea salt if desired.