Preheat oven to 350°. Grease 3 (8-inch) round cake pans; line bottoms with parchment paper, and grease and flour paper.
Stir together milk and vanilla.
Beat butter at medium speed with a heavy-duty electric stand mixer until creamy; gradually add sugar, beating until light and fluffy. Sift together flour and baking powder; add to butter mixture alternately with milk mixture, beginning and ending with flour mixture. Beat at low speed just until blended after each addition.
Beat egg whites at medium speed until stiff peaks form; gently fold into batter. Pour batter into prepared pans.
Bake at 350° for 20 to 23 minutes or until a wooden pick inserted in center comes out clean. Cool in pans on wire racks 10 minutes. Remove from pans to wire racks; discard parchment paper. Cool completely (about 40 minutes).
Spread Vanilla Buttercream Frosting between layers (about 1 cup per layer) and on top and sides of cake.