01 - Melt the 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, about 5 to 7 minutes. Remove from heat immediately 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 mixing 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. Gently fold everything together using a spatula until just combined, being careful not to overmix.
06 - Pour the batter into the prepared cake pan and use a spatula to smooth the top into an even layer.
07 - Bake 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 turn it out onto 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 syrup to cook undisturbed until it turns a deep amber color, about 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 the sauce 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 fresh banana slices, chopped toasted nuts, or an extra drizzle of caramel sauce as desired.