01 - Preheat oven to 350°F. Grease a 9-inch round cake pan and line the base with parchment paper.
02 - Melt 1 cup of unsalted butter in a small saucepan over medium heat. Cook, stirring, until foaming subsides and brown specks appear, releasing a nutty aroma. Remove from heat and set aside to cool slightly.
03 - In a mixing 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. Stir in mashed bananas and vanilla extract.
05 - Fold the dry mixture into the wet ingredients. Add sour cream and mix gently until batter is just combined.
06 - Pour the batter into the prepared pan. Smooth the surface and bake for 40 to 45 minutes, or until a toothpick inserted in the center emerges clean. Let cool completely on a wire rack.
07 - Heat 1 cup granulated sugar in a dry saucepan over medium heat, stirring constantly until melted and amber in color. Carefully whisk in 6 tablespoons cubed unsalted butter until incorporated. Gradually add 1/2 cup heavy cream; mixture will bubble vigorously. Stir in flaky sea salt. Let cool to room temperature.
08 - Beat 1/2 cup softened unsalted butter until light and creamy. Gradually mix in 2 cups powdered sugar and 1/2 cup cooled salted caramel sauce. Add 2 to 3 tablespoons milk, a little at a time, until desired consistency is reached.
09 - Once the cake is cool, frost with caramel frosting or drizzle generously with salted caramel sauce. Finish with sea salt flakes, if desired.