Preheat oven to 350 degrees. Thoroughly grease and flour baking pans.
In a large bowl, cream butter and 1 1/2 cup sugar. Separately sift together the sifted caked flour, baking powder and salt, and in a small bowl, combine milk and vanilla. Add the dry and wet mixes alternately to the large bowl, and blend thoroughly.
With an electric mixer on low speed (these are her instructions, but I’ll admit to impatience and using high), beat 2/3 cup egg white until stiff, not dry.
Add gradually to the main mixture in the large bowl a 1/4 cup of sugar, beating constantly until stiff. Fold egg whites into batter.
Measure batter by cupful into well-greased and floured cake pans. (I used one 9 X 13 and one 8 X 8 and then measured them after cooled to make them three same-sized layers.)
Bake at 350 degrees for about 30 minutes, checking towards the end. Cake is done when a toothpick can be inserted in the center and come out clean.