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

In a large mixing bowl, whisk the cooled brown butter with 1 cup of granulated sugar until well combined. Crack in the eggs one at a time, mixing thoroughly after each addition. Stir in 2½ teaspoons of pure vanilla extract, then gently fold in 2 cups of cake flour, 1 teaspoon of baking powder, and a pinch of salt until just combined.

Incorporate ½ cup of sour cream and ¼ cup of whole milk into the batter. Mix until the batter is smooth and creamy, ensuring there are no lumps.

Preheat your oven to 350°F (175°C) while you butter a 9-inch round cake pan. Pour the batter into the prepared pan, spreading it evenly. Sprinkle ¼ cup of granulated sugar over the top. Bake for 35–40 minutes, or until a toothpick inserted comes out with a few moist crumbs.

Let the French Butter Cake cool in the pan for about 10 minutes before transferring it to a wire rack to cool completely. Dust with powdered sugar before serving, and pair with fresh berries or whipped cream.