In a large bowl, cream the butter and sugars until light and fluffy. Add egg yolks and vanilla, one at a time. Combine the flour, baking powder, salt, and baking soda in a small bowl. Once the dry ingredients are well mixed, gradually add to the creamed mixture.
Divide the dough into 3 equal portions; tint each a different color. Roll each portion of dough between 2 sheets of waxed paper into a 9 inch by 5 inch rectangle. Freeze for 10 minutes.
Cut each rectangle in half lengthwise. Lightly brush to top of one rectangle with egg white. Top with another colored dough. Repeat with remaining 4 dough portions so you have six alternating layers of colored dough. Press together lightly. Cut in half lengthwise again. Wrap each with plastic wrap and refrigerate for several hours or overnight.
Preheat oven to 350 degrees. Unwrap the dough and cut into 1/8 inch slices. Place 2 inches apart ungreased baking sheets. Bake at 350 degrees for 8-10 minutes. Cool for 1-2 minutes before removing from pans to wire racks to cool completely.