In a medium bowl stir together all-purpose flour, baking soda, and cream of tartar; set aside.
In a large bowl, beat butter with electric mixer on medium to high speed for 30 seconds. Add powdered sugar. Beat until combined, scraping sides of bowl occasionally. Beat in egg, orange peel, and orange extract. Beat in as much of the flour as you can with the mixer. Stir in any remaining flour. Stir in cherries and pistachios.
Divide dough in half. Shape each half into an 11-inch-long roll. Wrap dough in plastic wrap or waxed paper. Chill in the refrigerator for 2 to 48 hours or in the freezer for at least 1 hour or until firm enough to slice.
Cut rolls into 1/2-inch slices; place 2 inches apart on greased cookie sheets. Bake in a 375 degree F oven for 8 to 10 minutes or until the edges are firm and slightly golden brown. Transfer cookies to wire racks; let cool.
If you like, decorate cookies. Drizzle with white chocolate or sprinkle with powdered sugar.