01 - Preheat oven to 350°F. Grease and line a 9-inch round cake pan with parchment paper.
02 - Melt 1 cup of butter in a small saucepan over medium heat. Cook, stirring frequently, until butter foams and brown flecks form with a nutty aroma. Remove from heat and let cool slightly.
03 - In a medium 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 brown sugar until combined. Beat in eggs one at a time. Stir in mashed bananas and vanilla extract.
05 - Gently fold the dry ingredients into the wet mixture. Add sour cream and stir just until incorporated.
06 - Pour batter into 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 completely on a wire cooling rack.
07 - In a dry saucepan over medium heat, cook sugar, stirring constantly, until dissolved and amber in color. Carefully whisk in butter until melted, then slowly add heavy cream, stirring; mixture will bubble. Stir in flaky sea salt and let cool to room temperature.
08 - Using a mixer, beat softened butter until creamy. Gradually introduce powdered sugar and 1/2 cup caramel sauce. Mix in milk, a tablespoon at a time, to achieve desired consistency.
09 - Once cake is completely cool, either frost with caramel frosting or generously drizzle with salted caramel sauce. For garnish, sprinkle additional flaky sea salt if desired.