01 - In a light-colored saucepan, melt 1/2 cup unsalted 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. Immediately remove from heat and transfer to a heatproof bowl. Allow to cool for 10 minutes.
02 - Preheat the 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 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 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 or Greek yogurt 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 cake pan and smooth the top with a spatula to ensure an even surface.
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 transfer to a wire rack to cool completely.
08 - In a clean saucepan, combine 1 cup granulated sugar and 1/4 cup 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 6 tablespoons cubed butter — the mixture will bubble vigorously. Whisk until the butter melts, then slowly pour in 1/2 cup 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 fresh banana slices, chopped toasted nuts, and an extra drizzle of caramel sauce as desired.