Preheat the oven to 350 degrees. Brush the cake pans with butter (see below for pan sizes and batter amounts). Line with parchment paper; butter parchment, and dust with flour, tapping out any excess. Set aside.
Whisk together the flour, baking powder, and salt in a medium bowl; set aside. Stir the milk and vanilla to combine; set aside. Beat the butter in a mixer bowl until pale and fluffy, about 3 minutes. Add the sugar in a steady stream; mix until pale and fluffy, about 3 minutes.
Reduce the speed to low. Add the flour mixture in 3 additions, alternating with the milk mixture, and beginning and ending with the flour; mix just until combined.
In a clean mixing bowl, whisk the egg whites just until stiff peaks form. Fold one-third of the egg whites into the batter to lighten. Gently fold in the remaining whites in two batches. Divide the batter among the prepared pans; smooth the tops with an offset spatula. Firmly tap the pans on a work surface to release any air bubbles.
Bake until a cake tester inserted into the centers comes out clean and the tops are springy to the touch (see below for baking times). Let cool in the pans on wire racks for 15 minutes. Invert the cakes onto the racks. Remove the parchment; reinvert and cool completely.
Approximate batter amounts and baking times for 3-inch-deep round pans: 6-inch layer: 3 cups batter, 30 minutes; 8-inch: 5 cups, 40 minutes; 10-inch: 8 cups, 1 hour 5 minutes; 12-inch: 11 cups, 1 hour 15 minutes.
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