01 - Preheat oven to 350°F. Grease and line a 9-inch round cake pan with parchment paper.
02 - Melt 1 cup unsalted butter in a small saucepan over medium heat, stirring frequently, until the butter foams and brown flecks form. Remove from heat and allow to cool slightly.
03 - In a medium bowl, whisk together all-purpose flour, baking soda, baking powder, and salt until evenly combined.
04 - In a large bowl, combine cooled brown butter with granulated sugar and light brown sugar. Blend in eggs one at a time. Stir in mashed bananas and vanilla extract until incorporated.
05 - Fold the flour mixture into the wet ingredients. Add sour cream and gently mix just until the batter is fully combined.
06 - Pour batter into the prepared pan. Smooth the surface and bake 40 to 45 minutes, or until a toothpick inserted in the center emerges clean. Cool entirely on a wire rack before frosting or serving.
07 - In a dry saucepan over medium heat, melt granulated sugar, stirring constantly, until fluid and deeply amber. Whisk in cubed butter until melted, then carefully add heavy cream, stirring until mixture is smooth and combined. Mix in flaky sea salt. Allow to cool to room temperature.
08 - Beat softened butter until creamy, then gradually add powdered sugar and 1/2 cup cooled salted caramel sauce. Adjust consistency with milk as needed until light and spreadable.
09 - Once cake is completely cool, frost the top with caramel frosting or generously drizzle with salted caramel sauce. Finish with a sprinkle of sea salt flakes, if desired.