Cream the softened butter and sugar until light and fluffy with an electric mixer. Mix in the egg yolk and vanilla.
Combine the flour, baking powder, and salt in a small bowl. Add about half of the flour and mix until incorporated with the mixer. Mix in the remaining flour and stir until combined with a wooden spoon. The dough will be very thick.
Divide the dough in half and add the cocoa powder to half of the cookie dough. Mix until the cocoa powder is even throughout; kneading if necessary.
Shape the chocolate dough into a rectangular log that is 6 inches (15 cm) in length and 2 inches (5 cm) in height. Repeat this process with the vanilla dough so you have two uniform rectangular logs of dough. Try to keep the edges as square as possible. Wrap each dough gently with plastic wrap and refrigerate for about an hour or freeze for 20 minutes to firm up.
Once firm, unwrap each rectangle of dough and cut lengthwise into 3 even slices using a large, sharp knife. Then, cut each piece into 3 additional long strips for a total of 9 strips of each color. See instructions below for pictures of this process.
Brush the strips with the whisked egg white and stack them in threes, with alternating colors to resemble a checkboard. Each log will contain a total of 9 pieces and be the same shape as the original logs. Once formed, wrap the logs in plastic wrap and refrigerate for about an hour or freeze for 20 minutes for easier slicing.
Once ready to bake, preheat the oven to 375° F. Line your cookie sheets with parchment paper. Unwrap the logs of dough and cut into ¼ inch slices. Place the cookies on the parchment lined cookie sheet and bake in the preheated oven for 8-10 minutes, or until the cookies are firm and lightly golden on the edges. Cool on the baking sheet for about a minute before transferring to a cooling rack.