Preheat oven to 325 degrees F (165 degrees C). Grease and flour a 9x13 inch pan. Sift together the flour, baking powder and baking soda; set aside. In a measuring cup combine the milk and vanilla; set aside.
In a large bowl, cream together the butter and white, confectioners' and brown sugars until light and fluffy. Beat in the egg whites, then the yolk. Beat in the flour mixture alternately with the milk, mixing just until incorporated.
Spread batter evenly into prepared pan. Bake for 50 to 60 minutes in the preheated oven, until a toothpick inserted in the center of the cake comes out clean. Cool in the pan for 10 minutes before inverting onto a wire rack to cool completely.