Heat the oven to 350°F. Butter and flour three 9-inch round cake pans and line the bottoms with waxed or parchment paper. Butter and flour the paper. Sift together the flour and the baking powder. Set aside. Combine the eggs and milk in a liquid measuring cup. Set aside.
In the bowl of a heavy-duty mixer fitted with the paddle attachment, cream the butter and sugar on medium speed until light and fluffy, about 3 minutes. Add the flour mixture and the milk mixture to the butter-sugar mixture, alternating between the dry and wet ingredients in three portions, starting and ending with the dry ingredients. Pour into the prepared pans.
Bake until a cake tester inserted into the center of each cake comes out clean and the cakes start pulling away from the sides of the pans, about 25 minutes. Remove to a rack to cool slightly. Invert onto the rack to cool completely.
To assemble the cake, place one cake layer on a cardboard cake round. Spread with the still-warm frosting. Repeat with remaining layers, placing the final layer bottom side up. Working quickly, use a small offset spatula to spread the icing gently around the cake. Let stand for 2 hours to allow the icing to set before serving. Store in an airtight container for up to 1 week.