In a small saucepan, melt 1 cup of unsalted butter over medium heat. Stir continuously until the butter turns golden brown and emits a nutty aroma, about 5-7 minutes. Remove from the heat and let it cool for about 10 minutes.

In a large mixing bowl, combine the cooled brown butter with 1 cup of granulated sugar and 1/2 cup of light brown sugar. Using an electric mixer, beat the mixture on medium speed until smooth, about 2 minutes. Add in the room temperature eggs, one at a time, mixing well after each addition, followed by 1 tablespoon of vanilla extract.

In a separate bowl, whisk together 2 cups of cake flour, 2 teaspoons of baking powder, and 1/2 teaspoon of salt. Gradually add the dry mixture to the wet ingredients, folding gently with a spatula until just combined.

Incorporate 1/2 cup of sour cream and 1/4 cup of milk into the batter, mixing until smooth.

Preheat your oven to 350°F (175°C). Grease and flour a 9-inch round cake pan thoroughly. Pour your batter into the prepared pan, smoothing the top with a spatula.

Place the cake pan in your preheated oven and bake for 35-40 minutes. Keep an eye on it, and when it’s golden brown and a toothpick inserted into the center comes out with moist crumbs, it’s done.

Once baked, allow the cake to cool in the pan for about 10 minutes. Then, gently transfer it to a wire rack to cool completely.