Preheat the oven to 350 degrees F. Butter the bottom and sides of two 9-by-2-inch round cake pans. Line the bottoms with parchment paper and butter the parchment. Flour the pans and tap out the excess.
Whisk the flour, baking powder and salt in a medium bowl. Beat the butter, 1 cup sugar and the vanilla in a large bowl with a mixer on low speed until combined. Increase the speed to medium high and beat until pale and fluffy, 5 to 7 minutes. Reduce the speed to low; beat in the flour mixture in two batches, alternating with the milk. Increase the speed to medium high and beat until smooth and glossy, 30 seconds to 1 minute.
Beat the egg whites in a clean bowl with a mixer on medium-high speed until foamy. Gradually beat in the remaining 1/3 cup sugar until stiff glossy peaks form, about 5 minutes. Fold half the egg whites into the batter, then fold in the rest. Divide the batter between the pans.
Bake the cakes until lightly browned, beginning to pull from the sides and a toothpick inserted into the centers comes out clean, 25 to 35 minutes. Transfer to a rack and let cool 15 minutes in the pans. Loosen the edges with a knife and turn out the cakes onto the rack; remove the parchment and let cool completely.