In a large mixing bowl, combine butter, sugar, egg, and the extracts. Beat at medium speed until creamy. Add flour, baking powder, and salt. Beat until well-blended. Divide dough into thirds. Add chopped maraschino cherries and red food coloring to one of the thirds. Mix well by hand.
(Note: I found it necessary to add a spoonful of flour to the cherry dough as it was very wet.) Add pistachio nuts (if using), pistachio pudding mix, and green food coloring to another third of the dough. Mix well by hand.
Shape each third of dough on separate sheets of parchment paper into logs approximately 1 1/2 inch in diameter and approximately 16 inches long. Flatten each log into a rectangle approximately 4 inches wide. You will have 3 rectangles of 4×16. Layer the dough on a single piece of plastic food wrap by lining up the rectangles and flipping them one on top of the other. Start with pistachio as the bottom layer, flip the plain layer on top of pistachio, and flip the maraschino cherry layer on top of the plain layer.
At this point you have one slab of dough, colored green, white and red. Next cut the dough in half so each piece is 2×16-inches. Place one half of dough on top of the other half, creating two sets of green, white and red patterns. Trim any long edges so you have a nice, clean edge all around your dough. Wrap securely in plastic wrap and refrigerate until firm, about 2 hours.
Note: To make it easier to handle, I would recommend cutting the finished dough log into two equal sized pieces of 8 inches, as 16 inches is very long and hard to handle.
When the dough is chilled well, preheat your oven to 375-f degrees. Cut the rectangle of dough into 1/4 inch slices. Place slices 1 inch apart on an ungreased cookie sheets. Bake for 12-14 minutes, until the dough no longer looks “wet.” (Note: my cookie edges never turned brown.) Cool completely. The cookies are crisp after cooling but will soften up a bit if you store them in a Tupperware overnight or in a Ziploc bag.