Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.

Whisk together the flour, baking powder, baking soda, and salt. Set aside.

Using an electric mixer on medium speed, beat the butter and brown sugar until light and fluffy. Add the eggs, one at a time, mixing well after each addition. Mix in the vanilla.

Reduce mixer speed to low. Add flour mixture in 3 portions, alternating with 2 portions of the sour cream. Mix just until combined.

Transfer to the prepared pan and spread evenly.

Bake 30-35 minutes, or until a pick inserted into the center comes out clean. Cool the cake in the pan on a wire rack for 10 minutes. Then, remove the cake from pan to a wire rack to cool completely.

Place the butter in a medium heavy saucepan over medium heat. Cook the butter until melted. Continue cooking, whisking often to prevent burning. The butter will begin to foam, brown specks will begin to form at the bottom of the pan, and the butter will have a nutty aroma.

Remove from the heat. Transfer the butter to a large heatproof bowl to cool to room temperature.

Add 2 tablespoons of milk and the vanilla to the cooled butter. Add about 1/4 cup of the confectioners' sugar. Beat the mixture until blended. Gradually add the remaining confectioners' sugar and mix until blended and smooth. If necessary, add more milk, 1 tablespoon at a time, and mix until the icing is the desired consistency.

Spread the icing over the top of the cooled cake. Serve immediately or refrigerate until ready to serve. If refrigerating, allow to sit at room temperature for about 30 minutes before serving.