In a small saucepan, melt the unsalted butter over medium heat, stirring frequently until it turns golden brown and smells nutty, about 5-7 minutes. Let it cool for about 10 minutes afterward.

In a large mixing bowl, whisk the cooled brown butter with granulated sugar until well combined. Add the large eggs one at a time, beating well after each addition.

Gently stir in the vanilla extract, cake flour, baking powder, and salt. Mix until just combined; avoid overmixing to keep the texture tender.

Add the sour cream and milk to the batter, mixing until it's smooth and creamy, ensuring no lumps remain.

Pour the batter into a well-buttered 9-inch cake pan. Smooth the top, sprinkle with a bit of sugar, and bake in a preheated oven at 350°F for 35-40 minutes, or until a toothpick comes out with a few moist crumbs.

Remove the cake from the oven and let it cool in the pan for about 10 minutes. Then, transfer it to a wire rack to cool completely before serving.

Optional: Top with fresh berries for a delightful contrast!