Preheat oven to 350 degrees F.
In the bowl of an electric mixer, cream butter and sugar until well blended, 1 to 2 minutes. Add vanilla, lemon extract, and salt. With mixer on low speed, gradually add flour, scraping down sides of bowl.
Turn dough out onto 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. Sprinkle cocoa powder over 1 half. Continue to knead until cocoa is fully incorporated. Using a rolling pin, shape each half of dough between plastic wrap into a 7 by 7-inch, 1/2-inch-thick square. With a sharp knife, slice each portion of dough into 9 (3/4-inch) strips; use a ruler to measure the precise 3/4-inch increments.
Whisk together egg and 1 tablespoon water. Cover work surface with plastic wrap. Place 3 strips of dough on plastic, alternating white and chocolate. Brush tops of strips and in between with egg wash. Press together gently. Form second and third layers by adhering alternate colors with egg wash and stacking to create the effect of a checkerboard. Roll into a log. Wrap log in plastic wrap. Repeat process for second log, reversing color pattern. Refrigerate dough for 30 minutes, or freeze for 15 minutes.
Slice each log into 1/4-inch-thick slices, and place on a silpat-lined baking sheet or parchment paper-lined baking sheet. Bake for 10 to 12 minutes. Let cool for 2 minutes, and transfer to a wire rack. Allow to cool completely.
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