Begin by placing 1 cup of unsalted butter in a small saucepan over medium heat. Allow it to melt completely, then continue cooking while stirring frequently until it turns a golden brown color and releases a nutty aroma, about 5-8 minutes. Remove from heat and let the browned butter cool for 10 minutes.

In a large mixing bowl, combine the cooled browned butter with 1 cup of granulated sugar. Whisk together until the mixture is well combined and slightly fluffy. Next, add 3 large eggs, one at a time, mixing thoroughly after each addition to ensure a smooth batter. Finally, stir in 2 teaspoons of vanilla extract for extra flavor.

In a separate bowl, sift together 2 cups of cake flour, 1 teaspoon of baking powder, and ½ teaspoon of salt. Gradually fold the dry ingredients into the wet mixture, being careful not to overmix; stop as soon as everything is just combined for a tender final texture in your French Butter Cake.

To enhance the richness of your batter, incorporate ½ cup of sour cream and ¼ cup of whole milk. Mix gently until the batter is smooth and has a lovely creamy consistency.

Carefully pour the prepared batter into a greased and floured 9-inch round cake pan. Spread it evenly and sprinkle an extra tablespoon of granulated sugar on top to create a deliciously crispy crust during baking.

Preheat your oven to 350°F (175°C). Place the cake pan in the oven and bake for 35-40 minutes, or until the top is golden brown and a toothpick inserted into the center comes out with a few moist crumbs.

Once baked, remove your French Butter Cake from the oven and let it cool in the pan for about 10 minutes. Carefully run a knife around the edges to loosen it, then invert the cake onto a wire rack to cool completely.