Combine the dough: In the bowl of an electric mixer, cream butter and sugar until well blended, 1 to 2 minutes. Add vanilla extract, lemon extract, and salt. With mixer on low speed, gradually add flour, scraping down the sides of bowl.
Knead the dough: Turn dough out onto a clean work surface; it will be loose and crumbly. 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. Put the other half back in the mixer and add the cocoa powder; mix until fully incorporated.
Shape the dough: Place each half of the kneaded dough between two sheets of parchment paper. Using a rolling pin, shape dough into two 7-inch squares, about ⅜ inch thick. Using a sharp knife and a ruler, slice each square into nine ¾-inch-wide strips.
Assemble the cookies: Whisk together the egg and 1 tablespoon water. Cover work surface with plastic wrap. Place three 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 colors to create a checkerboard effect. Wrap assembled log in plastic. Repeat process for second log, reversing color pattern. Refrigerate 30 minutes, or freeze 15 minutes.
Preheat the oven to 350℉.
Bake: Line two baking sheets with a Silpat baking mat or parchment paper. Slice each log into ¼-inch-thick slices; place cookies on baking sheets. Bake until done, 10 to 12 minutes. Remove baking sheets from oven, and let cookies cool 2 minutes. Transfer cookies to a wire rack to cool completely.