01 - In a light-colored saucepan, melt the butter 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. Remove from heat immediately and transfer to a bowl 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 the granulated sugar and 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 together using a spatula until just combined. Avoid overmixing to keep the cake tender.
06 - Pour the batter into the prepared pan and smooth the top with a spatula. Bake for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean. Let the cake cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.
07 - In a clean saucepan, combine the granulated sugar and water. Stir over medium heat just 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 remove from heat and add the cubed butter — the mixture will bubble vigorously. Whisk until the butter is fully melted, then slowly pour in the heavy cream while whisking continuously. Return to low heat and simmer for 1 to 2 minutes. Remove from heat and stir in the flaky sea salt. Let the sauce cool slightly before using.
08 - 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 extra drizzle of caramel as desired. Slice and serve.