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 brown flecks appear. Remove from heat and let cool slightly.
03 - In a medium bowl, whisk together all-purpose flour, baking soda, baking powder, and salt until combined.
04 - In a large bowl, mix cooled brown butter with granulated sugar and light brown sugar. Add eggs one at a time, beating well after each addition. Stir in mashed bananas and vanilla extract.
05 - Gradually fold the dry ingredients into the wet mixture. Add sour cream and gently mix until just combined; do not overmix.
06 - Transfer batter to the prepared pan. Smooth the surface and bake for 40 to 45 minutes, or until a toothpick inserted in the center comes out clean. Cool the cake completely on a wire rack.
07 - For caramel sauce, heat 1 cup sugar in a dry saucepan over medium heat, stirring until melted and amber in color. Carefully whisk in 6 tablespoons butter. Gradually add heavy cream as mixture bubbles, then stir in flaky sea salt. Allow to cool to room temperature.
08 - In a bowl, beat softened butter until creamy. Add powdered sugar gradually, then blend in 1/2 cup cooled salted caramel sauce. Mix in milk as needed until the frosting reaches a smooth, spreadable consistency.
09 - Once cake is completely cooled, frost with caramel frosting or drizzle generously with salted caramel sauce. Finish with extra sea salt flakes if desired.