In a small saucepan, melt the butter over medium heat, stirring frequently until it turns golden brown and develops a nutty aroma, about 5-7 minutes.

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

Gently fold in the cake flour, baking powder, and salt until just combined. Mix in the sour cream and milk until the batter is smooth.

Preheat your oven to 350°F (175°C) and prepare a 9-inch round cake pan by greasing it and dusting with flour. Pour the batter into the pan, smoothing the top.

Bake for 35-40 minutes, or until the top is golden brown and springs back lightly when touched. Check for doneness with a toothpick.

Let the cake cool in the pan for 10 minutes, then transfer it to a wire rack to cool completely.