Preheat oven to 350 degrees. Butter two each 5-, 7-, and 9-inch-diameter cake pans; line bottoms with parchment paper. Butter parchment; dust pans with flour, tapping out excess. Whisk flour, baking powder, and salt in a bowl. Mix milk and vanilla in a measuring cup.
Cream butter in a mixer bowl on medium-high speed until pale and fluffy, about 3 minutes. With mixer running, add sugar in a steady stream; beat until pale and fluffy, about 3 minutes more.
Reduce speed to low. Add flour mixture in three batches, alternating with milk mixture and beginning and ending with the flour; mix just until combined.
In a clean mixer bowl, whisk egg whites just until stiff peaks form. Use a rubber spatula to fold 1/3 of the whites into the batter to lighten. Gently fold in remaining whites in two batches. Divide batter among pans (see batter amounts, below); smooth tops. Firmly tap pans on a work surface to release any air bubbles.
Bake until a cake tester inserted into centers comes out clean and tops are springy to the touch(see times, below). Let cool in pans on wire racks for 15 minutes. Turn out cakes onto racks, and remove parchment; then reinvert, and cool completely.
Batter amounts and cooking times: 9-inch tier: 4 1/2 cups batter per 2-inch-high pan (9 cups total), bake about 40 minutes; 7-inch: 3 cups batter per pan (6 cups total), 30 minutes; 5-inch: 1 1/2 cups batter per pan (3 cups total), 22 minutes.
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