Melt brown butter in a small saucepan over medium heat, stirring constantly until golden brown (about 5-7 minutes). Remove from heat and cool for 10 minutes.

In a large bowl, mix cooled brown butter with sugar until well-blended. Add eggs one at a time, mixing after each, then stir in vanilla.

In another bowl, sift together cake flour, baking powder, and salt. Gradually fold into the wet mixture until just combined.

Gently fold in sour cream and milk until smooth.

Preheat oven to 350°F (175°C). Grease and flour a 9-inch round cake pan. Pour in the batter and sprinkle with sugar if desired.

Bake for 35-40 minutes until golden brown and a toothpick comes out clean.

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