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. Cook, stirring, until butter foams, develops brown flecks, and gives a nutty aroma. Remove from heat and allow to cool slightly.
03 - In a medium bowl, whisk together all-purpose flour, baking soda, baking powder, and salt.
04 - In a large bowl, blend cooled brown butter with granulated sugar and light brown sugar. Beat in eggs one at a time. Stir in mashed bananas and pure vanilla extract.
05 - Fold dry ingredients into the wet mixture until just combined. Add sour cream and gently mix to incorporate.
06 - Transfer batter to prepared pan. Smooth the surface and bake for 40 to 45 minutes, or until a toothpick inserted in the center emerges clean. Cool completely on a wire rack.
07 - To make caramel, heat 1 cup granulated sugar in a dry saucepan over medium heat, stirring constantly until melted and amber colored. Carefully whisk in butter until fully melted, then slowly add heavy cream (mixture will bubble). Stir in flaky sea salt and allow to cool to room temperature.
08 - For frosting, beat softened butter until creamy. Gradually blend in powdered sugar and 1/2 cup cooled caramel sauce. Add milk gradually to reach desired spreading consistency.
09 - Once cake is fully cooled, apply caramel frosting or generously drizzle with salted caramel sauce. Decorate with additional sea salt flakes if desired.