Preheat oven to 350 degrees. Brush two 8-by-8-by-2-inch square cake pans with butter. Line pans with parchment; butter parchment and dust with flour, tapping out any excess. Set aside.
Whisk together flour, baking powder, and salt in a large bowl; set aside. Stir the milk and vanilla to combine; set aside. Beat the butter in a mixer bowl on medium high until pale and fluffy, about 3 minutes. Add 2 1/4 cups of sugar in a steady stream and mix until pale and fluffy, about 3 minutes.
Reduce speed to low. Add the flour mixture in 3 additions, alternating with the milk mixture, beginning and ending with the flour. Mix until just combined.
In a clean mixing bowl, beat the egg whites until frothy. Slowly add remaining 3 tablespoons sugar and beat until stiff peaks form. Fold one-third of the egg whites into batter to lighten. Gently fold in remaining egg whites in two additions.
Divide batter between prepared pans; smooth the tops with an offset spatula. Firmly tap the pans on a work surface to release any bubbles.
Bake until a tester comes out clean, about 55 minutes. Cool in pans on a wire rack about 15 minutes. Invert cakes onto wire rack. Remove parchment, reinvert, and cool completely.