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 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 2 cups all-purpose flour, 1 teaspoon baking powder, 1/2 teaspoon baking soda, and 1/2 teaspoon salt until evenly distributed.
04 - In a large bowl, whisk the cooled brown butter with 1/2 cup granulated sugar and 1/2 cup packed light brown sugar until smooth. Beat in 2 large eggs one at a time, mixing well after each addition. Stir in 1 teaspoon vanilla extract, the mashed bananas (about 1 1/4 cups), and 1/2 cup sour cream or Greek yogurt 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. 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 1 cup granulated sugar and 1/4 cup 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, 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 1 to 1 1/2 teaspoons flaky sea salt. 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 additional drizzle of caramel as desired.