01 - Melt unsalted butter in a light-colored saucepan over medium heat. Continue cooking, swirling the pan occasionally, until the butter foams and golden-brown specks form on the bottom, approximately 5 to 7 minutes. Immediately remove from heat and transfer to a heatproof bowl. Allow to cool for 10 minutes.
02 - Preheat the oven to 350°F. Grease a 9-inch round or square cake pan and line the bottom with parchment paper.
03 - In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt until evenly distributed.
04 - In a large bowl, whisk the cooled brown butter with granulated sugar and packed light brown sugar until smooth. Beat in the eggs one at a time, mixing well after each addition. Stir in the vanilla extract, mashed bananas, and sour cream until fully incorporated.
05 - Add the dry ingredient mixture to the wet ingredients and gently fold with a spatula until just combined. Do not overmix.
06 - Transfer the batter into the prepared cake pan and smooth the top with a spatula to ensure an even surface.
07 - Bake on the center rack for 40 to 45 minutes, or until a toothpick inserted into the center of the cake comes out clean. Let the cake cool in the pan for 10 minutes, then invert onto a wire rack to cool completely.
08 - In a clean, dry saucepan, combine granulated sugar and water. Stir over medium heat only until the sugar dissolves, then stop stirring. Allow the mixture to cook undisturbed until it turns a deep amber color, approximately 6 to 8 minutes. Immediately add the cubed butter, whisking vigorously as it will bubble vigorously. Once the butter is fully melted, slowly pour in the heavy cream while whisking continuously. Simmer for 1 to 2 minutes, then remove from heat and stir in the flaky sea salt. Let the sauce cool slightly before using.
09 - Once the cake has cooled completely, pour the warm salted caramel sauce evenly over the top. Garnish with banana slices, chopped toasted nuts, or an additional drizzle of caramel sauce as desired.