Heat the oven to 350 degrees. Butter and flour two 8-by-3-inch cake pans. In the bowl of an electric mixer fitted with the paddle attachment, on medium speed, cream together butter, sugar, and vanilla until light and fluffy, about 3 minutes.
Sift together flour and baking powder. With the mixer set on low speed, add the dry ingredients to the butter mixture, alternating with milk, starting and ending with flour; scrape down sides twice.
In the bowl of an electric mixer fitted with the whisk attachment, whisk egg whites with pinch of salt on high speed until stiff peaks form, 3 to 5 minutes. Whisk in a large spoonful of the beaten whites into the batter to lighten the mixture. Fold the lightened batter into the remaining egg whites. Divide the batter equally among the prepared pans. Bake until golden brown and a cake tester inserted into the center comes clean, about 1 hour 5 minutes. Remove the pans to a rack to cool for 5 minutes. Invert cakes onto wire rack to cool completely.
If cakes are not level, use a serrated knife to trim tops off. Carefully slice each cake horizontally into three equal layers, about 3/4 inch thick. (You will have six layers.) To assemble, place one sliced layer on an 8-inch cardboard cake round. Spread with 1/2 cup frosting. Repeat with remaining layers, placing the final layer bottom side up. Cover cake with remaining frosting. Serve immediately.