01 - Preheat 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. Swirl the pan frequently as it foams and sputters, watching carefully until the milk solids turn golden brown and release a nutty, toasty aroma. Immediately pour into a heatproof bowl and let cool slightly.
03 - In a medium bowl, whisk together the 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 cooled brown butter. Stir thoroughly until the mixture is smooth and homogeneous.
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 pans, smoothing the tops with a spatula.
07 - Bake on the center rack for 35 to 40 minutes, or until a toothpick inserted into the center of each layer comes out clean. Let the cakes rest in their pans for 10 minutes, then invert onto wire racks to cool completely.
08 - 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 add the cubed butter, stirring until fully melted. Slowly whisk in the room-temperature cream, expecting vigorous bubbling and steam. Stir in the flaked sea salt and set aside to cool to room temperature.
09 - Beat the softened butter with an electric mixer on medium speed until light and creamy. Gradually add the sifted powdered sugar and a pinch of salt, mixing on low until incorporated. Increase speed and beat in 4 to 5 tablespoons of the prepared caramel sauce. Add milk or cream one 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 caramel sauce. Set the second layer on top. Apply frosting to the top and sides of the cake, then finish with additional caramel drizzle and a sprinkle of flaked sea salt if desired.