Set the oven to 350°F (177°C). Grease and line three 9-inch cake pans with parchment paper.
In a bowl, combine baking soda, baking powder, salt, and cake flour, salt.
In a mixer, beat sugar and butter at high speed until smooth (about 3 minutes). Scrape down sides occasionally.
Continue mixing, adding 3 whole eggs, 2 egg whites, and vanilla. Mix until combined.
At low speed, gradually add dry ingredients, then buttermilk, mixing until it is just combined. Stir by hand to eliminate lumps.
Divide the batter evenly between the pans and bake for 25 minutes. Cool the pans on a wire rack.
Beat butter until creamy. Gradually add confectioners' sugar, milk, vanilla, and salt, and beat until smooth. Adjust consistency if necessary.
Level cake layers using a serrated knife. Layer cakes with frosting, covering each top and sides.
Refrigerate the assembled cake for at least an hour before serving.