In a medium saucepan, melt 1 cup of unsalted butter over medium heat. Stir frequently until the butter turns golden brown and releases a nutty aroma, about 5–7 minutes.

In a large mixing bowl, whisk together the cooled brown butter and 1 cup of sugar until well combined. Add 2 large eggs, one at a time, mixing thoroughly after each addition. Stir in 1 teaspoon of vanilla extract.

In a separate bowl, sift together 2 cups of cake flour, 2 teaspoons of baking powder, and a pinch of salt. Gradually add this dry mixture to your wet ingredients, folding gently until just combined.

Gently incorporate 1 cup of sour cream and ¼ cup of milk into the batter. Mix until the batter is smooth and velvety.

Preheat your oven to 350°F (175°C). Grease and flour a 9-inch round cake pan. Pour the batter into the pan, smoothing the top gently. Sprinkle a tablespoon of sugar over the top before baking.

Place the prepared cake pan in the preheated oven and bake for 35–40 minutes until golden brown and a toothpick inserted in the center comes out with moist crumbs.

Once baked, remove the cake from the oven and let it cool in the pan for about 10 minutes. Transfer it to a wire rack to cool completely.