01 - Melt butter in a light-colored saucepan over medium heat. Continue cooking, swirling occasionally, until foaming subsides and golden brown specks form on the bottom, about 5 to 7 minutes. Immediately remove from heat and transfer to a bowl. Let cool for 10 minutes.
02 - Preheat oven to 350°F. Grease and line a 9-inch round or square cake pan with parchment paper.
03 - In a medium bowl, whisk together the flour, baking powder, baking soda, and salt until evenly distributed.
04 - In a large bowl, whisk the cooled brown butter with granulated sugar and 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 - 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 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 — it 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 sea salt. Let 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, and additional caramel as desired.