In a saucepan over medium heat, melt the unsalted butter, stirring frequently until it becomes foamy and turns a rich golden brown, about 5 minutes. Remove from heat and cool for 10 minutes.

In a mixing bowl, whisk together the cooled brown butter and sugar until well combined. Add eggs one at a time, mixing thoroughly after each addition, then incorporate vanilla extract.

Gradually add cake flour, baking powder, and salt to the wet mixture and gently fold the ingredients until just combined.

Gently mix in sour cream and milk into the batter until smooth, ensuring there are no lumps.

Grease a 9-inch round cake pan, pour the batter into the pan, and sprinkle sugar over the top.

Bake in a preheated oven at 350°F (175°C) for 35-40 minutes until golden brown. Cool for 10 minutes before transferring to a wire rack.