01 - Preheat the oven to 350°F. Grease and flour two 8-inch round cake pans, lining the bottoms with parchment paper for easy release.
02 - Melt 3/4 cup unsalted butter in a light-colored saucepan over medium heat. Swirl the pan frequently as the butter foams and begins to turn a deep golden brown with a fragrant, nutty aroma, approximately 5 to 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 mixing bowl, combine the mashed bananas, granulated sugar, brown sugar, eggs, buttermilk, vanilla extract, and the cooled brown butter. Stir until the mixture is smooth and thoroughly incorporated.
05 - Gradually fold the flour mixture into the wet ingredients using a spatula, mixing just until no dry streaks remain. Avoid overmixing to keep the cake tender.
06 - Divide the batter evenly between the two prepared cake pans, smoothing the tops. 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 invert onto wire racks to cool completely before assembling.
08 - In a medium saucepan, combine 1 cup granulated sugar and 1/4 cup water. Heat over medium heat without stirring, gently swirling the pan occasionally, until the sugar 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, taking caution as the mixture will bubble vigorously. Stir in the flaked sea salt and let the sauce cool to room temperature.
09 - Using a hand mixer or stand mixer, beat the softened butter on medium speed until light and creamy. Gradually add the sifted powdered sugar and a pinch of salt, mixing on low until incorporated. Beat in 4 to 5 tablespoons of the prepared salted caramel sauce. Add milk or cream, 1 tablespoon at a time, until the frosting reaches a smooth, spreadable consistency.
10 - Place one cake layer on a serving plate. Spread an even layer of caramel frosting over the top and drizzle generously with salted caramel sauce. Position the second cake layer on top. Frost the top and sides of the entire cake, then drizzle with additional caramel sauce. Finish with a sprinkling of flaked sea salt if desired.