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. Stir constantly as it foams and develops brown flecks. When the butter smells nutty and golden, remove from heat and let it cool slightly.
03 - In a medium bowl, whisk together all-purpose flour, baking soda, baking powder, and salt until well blended.
04 - In a large bowl, combine cooled brown butter, granulated sugar, and light brown sugar. Beat in eggs one at a time. Mix in mashed bananas and vanilla extract.
05 - Fold the dry mixture into the wet batter gently. Add sour cream and stir only until just incorporated. Do not overmix.
06 - Transfer the batter to the prepared pan and smooth the surface. Bake for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean. Cool cake completely on a wire rack.
07 - In a dry saucepan over medium heat, melt 1 cup granulated sugar, stirring constantly, until completely liquefied and amber in color. Carefully whisk in cubed butter until fully melted. Slowly add heavy cream while stirring; the mixture will bubble. Stir in flaky sea salt. Allow sauce to cool to room temperature.
08 - In a mixing bowl, beat softened butter until creamy. Gradually add powdered sugar and 1/2 cup cooled salted caramel sauce. Add 2 to 3 tablespoons milk as necessary to achieve a smooth, spreadable consistency.
09 - Once cake is fully cool, frost with caramel frosting or generously drizzle with salted caramel sauce. Optionally, sprinkle with extra sea salt flakes for garnish.