In a medium saucepan over medium heat, add unsalted butter. Allow it to melt and cook until golden brown for about 5 minutes.

In a large mixing bowl, combine the cooled brown butter with granulated sugar. Whisk together until blended. Add eggs one at a time and mix smoothly.

Sift together cake flour, baking powder, and salt in a separate bowl. Gently fold this dry mixture into the butter-sugar-egg mixture using a spatula.

Pour in sour cream and milk to the batter, mixing until smooth, being careful not to overmix.

Grease a 9-inch round cake pan and pour the batter in, smoothing it out. Sprinkle remaining sugar on top.

Preheat oven to 350°F (175°C) and bake for 35-40 minutes until the top is golden brown and a toothpick comes out with moist crumbs.

Once baked, cool in the pan for 10 minutes, then transfer to a wire rack to cool completely before slicing.