Cream the butter and sugar until well blended. Add the vanilla extract and with mixer on low speed, gradually add flour, scraping down sides of the bowl.
Turn dough out onto a clean work surface. Knead dough by pushing small amounts away from you with the heel of your hand for 1 to 2 minutes. Divide the dough in half then sprinkle cocoa powder over one of the halves.
Knead until the cocoa has been fully incorporated.
Place each half of the kneaded dough between 2 sheets of cling film. Using a rolling pin, shape dough into two 7-inch squares, about 1/2 inch thick. Using a sharp knife and a ruler, slice each square into nine 3/4-inch-wide strips.
Whisk together the egg and 1 tablespoon water.
Cover work surface with plastic wrap. Place 3 strips of dough on plastic, alternating white and chocolate strips. Brush tops and in between the strips with egg wash. Gently press strips together. Repeat, forming second and third layers, alternating colours to create a checkerboard effect.
Wrap assembled log in plastic.
Repeat process for the second log, reversing the colour pattern.
Refrigerate 30 minutes, or freeze 15 minutes.
Preheat oven to 350 degrees. Line a baking sheet with parchment paper. Slice each log into 1/4-inch-thick slices; place on baking sheet. Bake until done, 10 to 12 minutes. Remove baking sheet from oven, and let the cookies cool 2 minutes. Transfer the cookies to a wire rack to cool completely.