01 - Preheat the oven to 350°F (175°C). Grease and flour two 8-inch (20 cm) round cake pans, then line the bottoms with parchment paper.
02 - Melt the 3/4 cup unsalted butter in a light-colored saucepan over medium heat. Swirl the pan frequently as the butter foams and eventually turns a deep golden brown with a fragrant, nutty aroma. Transfer immediately to 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 cooled brown butter. Stir until fully incorporated and smooth.
05 - Gradually fold the dry ingredient 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. Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean. Let the cakes rest in the pans for 10 minutes, then turn out onto wire racks to cool completely.
07 - While the cakes bake, combine the granulated sugar and water in a medium saucepan over medium heat. Do not stir; swirl gently if needed. Once the sugar reaches 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, watching for steam and splatter, then stir in the flaked sea salt. Set aside to cool to room temperature.
08 - 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, mixing on low speed until combined. Beat in 4 to 5 tablespoons of the prepared caramel sauce and 1 to 2 tablespoons of milk or cream until the frosting is smooth and spreadable.
09 - 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. Set the second layer on top and frost the top and sides of the entire cake. Finish with an additional drizzle of caramel sauce and a sprinkle of flaked sea salt if desired.