01 - Preheat the 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, stirring occasionally. Continue cooking until the butter foams and brown flecks form, giving off a nutty aroma. Remove from heat and allow to cool slightly.
03 - In a medium bowl, whisk together the all-purpose flour, baking soda, baking powder, and salt.
04 - In a large bowl, mix the cooled brown butter with granulated sugar and light brown sugar. Beat in the eggs one at a time. Stir in the mashed bananas and vanilla extract.
05 - Fold the dry ingredients into the wet mixture until mostly combined, then gently mix in the sour cream. Do not overmix.
06 - Transfer the batter to the prepared pan, smoothing the surface evenly. Bake for 40 to 45 minutes, or until a toothpick inserted in the center comes out clean. Allow cake to cool completely on a wire rack.
07 - In a dry saucepan over medium heat, melt the granulated sugar, stirring constantly until fully melted and amber in color. Carefully whisk in the cubed butter until incorporated. Slowly pour in the heavy cream while stirring (mixture will bubble). Stir in flaky sea salt. Cool to room temperature.
08 - Beat softened unsalted butter until creamy. Gradually add powdered sugar and 1/2 cup cooled salted caramel sauce. Mix in milk, a tablespoon at a time, until the frosting reaches desired consistency.
09 - Once the cake has cooled, spread with caramel frosting or drizzle generously with salted caramel sauce. Garnish with additional sea salt flakes, if desired.