In the bowl of an electric mixer fitted with the paddle attachment, 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 sides of bowl.
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 dough in half. Put the other half back in the mixer and add cocoa powder; mix until fully incorporated.
Place each half of the kneaded dough between two sheets of plastic. Using a rolling pin, shape dough into two 7-inch squares, about 3/8 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 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 oven to 350 degrees.
Line a baking sheet with a Silpat baking mat or 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 cookies cool 2 minutes. Transfer cookies to a wire rack to cool completely.