01 - Preheat oven to 350°F. Grease and line a 9-inch round cake pan with parchment paper.
02 - In a small saucepan, melt 1 cup unsalted butter over medium heat. Cook, stirring constantly, until the butter foams and brown flecks appear with a nutty aroma. Remove from heat and let cool slightly.
03 - In a medium bowl, whisk together all-purpose flour, baking soda, baking powder, and salt.
04 - In a large bowl, blend the cooled brown butter with granulated sugar and light brown sugar. Add eggs one at a time, mixing well after each addition. Stir in the mashed bananas and vanilla extract.
05 - Fold the dry mixture into the wet ingredients using a spatula. Add sour cream and mix gently until just combined. Avoid over-mixing.
06 - Transfer the batter to the prepared pan. Smooth the surface and bake for 40 to 45 minutes, or until a toothpick inserted in the center emerges clean. Allow to cool fully on a wire rack.
07 - In a dry saucepan over medium heat, warm granulated sugar, stirring constantly, until melted and a deep amber hue. Immediately whisk in the cubed butter until dissolved, then cautiously pour in the heavy cream (mixture will bubble). Stir in the sea salt and cool to room temperature.
08 - Beat softened butter in a bowl until creamy. Gradually incorporate powdered sugar and half a cup of cooled salted caramel sauce. Add milk, one tablespoon at a time, for desired spreading consistency.
09 - Once cake has cooled, frost with caramel frosting or distribute the salted caramel sauce generously over the surface. Garnish with additional sea salt flakes if desired.