01 - Preheat the oven to 350°F. Grease a 9-inch round cake pan with butter and line the base with parchment paper.
02 - In a small saucepan, melt 1 cup of unsalted butter over medium heat, stirring frequently until it foams and develops brown flecks with a nutty aroma. Remove from heat and allow to cool slightly.
03 - In a medium mixing bowl, whisk together the all-purpose flour, baking soda, baking powder, and salt until thoroughly blended.
04 - In a large bowl, combine the cooled brown butter with granulated sugar and brown sugar. Beat in the eggs one at a time, followed by mashed bananas and vanilla extract.
05 - Gently fold the dry mixture into the wet ingredients. Add sour cream and mix until just combined, ensuring not to overmix the batter.
06 - Pour batter into the prepared cake pan and smooth the surface. Bake for 40 to 45 minutes or until a toothpick inserted in the center emerges clean. Transfer to a wire rack and allow to cool completely in the pan.
07 - Heat granulated sugar in a dry saucepan over medium heat, stirring constantly until fully melted and amber in color. Carefully whisk in cubed unsalted butter until melted, then gradually pour in heavy cream while whisking as the mixture bubbles. Stir in flaky sea salt and allow sauce to cool to room temperature.
08 - With a hand or stand mixer, beat softened butter until creamy. Gradually add powdered sugar and 1/2 cup cooled salted caramel sauce. Adjust consistency with 2 to 3 tablespoons milk if necessary, beating until smooth and spreadable.
09 - Once the cake is fully cooled, spread with caramel frosting or drizzle generously with salted caramel sauce. Optionally, garnish with additional flaky sea salt.