In large bowl, combine sugar, 1 cup butter, 2 teaspoons rum extract and egg; beat until light and fluffy. Lightly spoon flour into measuring cup; level off. Add flour; mix well. With spoon, stir in 1/2 cup pistachios and cherries. Cover with plastic wrap; refrigerate 30 minutes for easierhandling.
Heat oven to 350°F. Divide dough into 6 pieces. On lightly floured surface, shape each piece of dough into rope, 24 inches long and 1/2 inch in diameter. Cut into 2-inch-long pieces; place 1 inch apart on ungreased cookie sheets.
Bake at 350°F. for 9 to 15 minutes or until bottoms are lightly browned. Immediately remove from cookie sheets; place on wire racks. Cool completely.
In medium bowl, combine powdered sugar, 3 tablespoons butter, 1/2 teaspoon rum extract and enough milk for desired frosting consistency; blend well. Spread over tops of cookies. Sprinkle with 1/3 cup pistachios before frosting is set.