Place the butter into a large saucepan and melt over medium heat. Allow it to cook until it begins to foam and sizzle, stirring occasionally. Continue cooking until the butter develops a rich, nutty aroma and turns a deep golden brown—this should take around 10–15 minutes.

Once browned, immediately remove from heat. Measure out 10 tablespoons of the brown butter and transfer it to a mixing bowl (or the bowl of a stand mixer) for the cake. Pour the remaining butter into a separate large bowl to be used later in the frosting.

Refrigerate both portions until they solidify slightly but remain soft enough to whip.

Preheat the oven to 350°F (175°C). Lightly grease a 9×9-inch metal baking pan and line the bottom with parchment paper.

In a medium bowl, whisk together the flour, baking powder, baking soda, and salt, then set aside.

In the bowl with the smaller portion of browned butter, beat with an electric mixer (or use a paddle attachment on a stand mixer) on high speed until the butter becomes pale and fluffy.

Add the granulated sugar and brown sugar and cream together with the butter for about 2 minutes, until well combined.

Add the eggs and vanilla extract, mixing on medium speed until the mixture is pale and smooth, about 1 minute.

Gradually add the dry ingredients and buttermilk in alternating portions, mixing on low speed until the batter is smooth and fully incorporated. Scrape down the sides of the bowl as needed.

Pour the batter into the prepared pan and spread it evenly. Bake for 35–38 minutes, or until a toothpick inserted into the center comes out clean.

Once baked, allow the cake to cool completely in the pan on a wire rack.

While the cake is baking, prepare the caramel sauce. Add the butter and brown sugar to a medium saucepan and bring to a gentle simmer over medium heat. Let it simmer for 5 minutes, stirring occasionally.

Reduce the heat to medium-low and carefully whisk in the heavy cream. Simmer the mixture for another 3–4 minutes until slightly thickened.

Remove from the heat, then stir in the vanilla extract and salt. Transfer the caramel to a bowl and let it cool to room temperature on the counter.

While the cake is cooling, whip the larger portion of browned butter in a mixing bowl using an electric mixer on high speed for 5–10 minutes, until it becomes pale and fluffy.

Add in the cold cream cheese and continue mixing on medium speed until the mixture becomes smooth and fluffy.

Sift in the powdered sugar, mixing on low speed at first until incorporated, then increase to high speed and beat for 1 more minute until the frosting is light and airy.

Once the cake is completely cool, carefully lift it out of the pan using the parchment paper and transfer it to a serving plate.

Use the end of a wooden spoon or a skewer to poke holes all over the surface of the cake.

Transfer the cooled caramel sauce to a piping bag and snip the tip. Pipe the caramel directly into the holes on top of the cake. If the caramel is too thick, warm it slightly or run the filled piping bag under hot water to soften.

Spread a thick, even layer of the brown butter cream cheese frosting over the top of the cake. Drizzle any remaining caramel over the frosting, then use the back of a spoon to swirl it gently into the frosting for a marbled effect.

Cut into 16 squares and serve. For an extra indulgent touch, drizzle additional caramel over each slice just before serving.