Heat oven to 350°F. Combine 2/3 cup butter and sugar in large mixer bowl. Beat at medium speed, scraping bowl often, until creamy (1 to 2 minutes). Add eggs and vanilla. Continue beating, scraping bowl often, until well mixed (1 minute).
Reduce speed to low. Beat, gradually adding flour, baking powder and salt alternately with milk and scraping bowl often, until well mixed (1 to 2 minutes).
Divide batter among 2 greased and floured 8 or 9-inch round cake pans. Bake for 25 to 30 minutes or until toothpick inserted in center comes out clean. Cool 5 minutes; remove from pans. Cool completely.
Meanwhile, melt 1/4 cup butter in 1-quart saucepan over medium-low heat until lightly browned (4 to 5 minutes). Combine browned butter, powdered sugar and vanilla in small mixer bowl. Beat at medium speed, scraping bowl often and gradually adding enough milk for desired spreading consistency.
Spread frosting between cooled cake layers, top and sides of cake.