01 - Preheat oven to 350°F. Grease and flour two 8-inch round cake pans, tapping out any excess flour.
02 - Melt 3/4 cup unsalted butter in a light-colored saucepan over medium heat, swirling frequently. Continue cooking until the butter turns golden brown and releases a nutty, toasted aroma, about 5 to 7 minutes. 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 bowl, combine the mashed bananas, granulated sugar, brown sugar, eggs, buttermilk, vanilla extract, and slightly 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 just combined. 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. Let the cakes cool in the pans for 10 minutes, then turn out 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 sugar to dissolve and cook until it turns a deep amber color. Remove from heat and carefully stir in the cubed butter until melted. Slowly whisk in the room-temperature heavy cream, taking caution as the mixture will bubble vigorously. Stir in the flaked sea salt and 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 until combined. Beat 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.
09 - Place the first cake layer on a serving plate. Spread a generous, even 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 entire cake with the remaining frosting. Finish with additional caramel sauce drizzled over the top and a sprinkle of flaked sea salt if desired.