In a large bowl of mixer with the paddle attachment, cream butter and sugar until well blended, 1 to 2 minutes. Add extracts and salt. With mixer on low speed, gradually add flour, scraping down sides of bowl.
Turn dough out onto a 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 3 pieces. Add green gel into one piece, Knead until gel has been fully Incorporated. Repeat with red gel in another piece. Leaving third piece plain.
Place each piece of the kneaded doughs between 2 sheets of plastic. Using a rolling pin, shape dough into three 7-inch squares, about 3/8 inch thick. Using a sharp knife and a ruler, slice each piece into 3/4 inch strips.
Whisk together the egg and 1 Tablespoon water. Cover work surface with plastic wrap. Place strips of dough on plastic alternating green, white and red. 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 and third log,reversing color patterns. Refrigerate 30 minutes or freeze 15 minutes.
Preheat oven to 350^. Line a baking sheet with parchment. Slice each log into 1/4 inch thick slices; place on baking sheet. Bake until done, 10-12 minutes. Remove baking pan from oven, and let cookies cool 2 minutes. Transfer cookies to a wire rack to cool completely.