Preheat the oven to 400 degrees F. Grease and flour three 9-inch pans lined with parchment paper.
Cream the butter and sugar in the bowl of a stand mixer on medium speed for 5 minutes, or until light and fluffy.
Scrape the sides and bottom of the bowl.
Add the eggs one at a time, beating well after each addition.
Scrape the sides and bottom of the bowl again.
Combine the cake flour, baking powder, and salt in a bowl and mix with a wire whisk.
Combine the milk and vanilla extract in a liquid measuring cup.
With the mixer on low, alternate adding the flour mixture and the milk mixture, beating until well blended.
Beat the batter for 7 minutes on medium speed.
Divide the batter evenly between the three prepared pans. Tap the pans on the counter to ensure there are no air bubbles.
Bake for 18 to 20 minutes. Test doneness by touching a cake top. If it springs back, it is done. If the indentation stays, it needs 2 to 3 minutes longer.
Cool the cakes for at least 10 minutes on a cooling rack before removing from the pans.
Melt the butter in a medium saucepan over medium heat.
Add the brown sugar and stir using a wire whisk until the butter is absorbed into the sugar. You will know the butter is absorbed because there will not be butter pieces floating on the top. Allow the mixture to bubble for about 2 minutes, stirring occasionally.
After 2 minutes, add the milk very carefully to the mixture to avoid splattering. Keep stirring and return the mixture to a full, rolling boil. Remove the mixture from the heat.
Add the mixture to the bowl of a stand mixer.
With the mixer on low speed and using the whisk attachment, gradually add the confectioners\' sugar (1 cup at a time), mixing well after each addition of sugar. Mix until smooth and glossy; the icing should pour easily from the bowl.
While the icing is still warm, frost the cooled cake layers: Spread icing between the cake layers and then on the top and sides of the stacked layers. The icing will set up like candy as it cools.
If the mixture hardens while frosting the cake, warm the icing in the microwave in 10-second intervals until the smooth velvety texture has returned.