Let the butter stand at room temperature for 30 minutes. In a bowl, stir together the all-purpose flour, baking soda, and cream of tartar; set aside.
In a large mixing bowl, beat butter and powdered sugar with electric mixer on medium to high speed until mixture is fluffy. Beat in the egg and orange flavoring. Stir in the cherries, finely chopped pistachios, and finely shredded orange peel.
Shape dough into two 11-inch-long rolls, each about 2-1/2 inches in diameter. Wrap in clear plastic wrap and chill in the freezer for at least 2 hours.
Cut rolls into 1/2-inch slices; place 2 inches apart on greased cookie sheets.
Bake the cookies in a 375 degree F oven for 8 to 10 minutes or until the edges are lightly browned. Transfer cookies to wire racks; cool completely.
Decorate the cookies with stripes of melted white baking bar or sprinkle with some powdered sugar, if you like. Makes about 80 cookies.
Shape dough into logs. Wrap in clear plastic wrap and chill in the refrigerator for 4 to 24 hours.