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, incorporating each fully before adding the next. Stir in the vanilla extract, mashed bananas, and sour cream until well combined.
05 - Add the dry ingredients to the wet mixture and gently fold with a spatula until just combined. Do not overmix.
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.
07 - Let the cake cool in the pan for 10 minutes, then transfer to a wire rack to cool completely.
08 - 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 add the cubed butter, which will bubble vigorously. Whisk until the butter melts, then 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. 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 as desired.