Preheat oven to 350°. Beat first 3 ingredients at medium speed with an electric mixer until fluffy. Add egg, beating until blended.
Combine flour and next 3 ingredients in a medium bowl; gradually add to butter mixture, beating just until blended. Shape dough into a disk. Cover and chill 10 minutes.
Place dough on a lightly floured surface, and roll to 1/8-inch thickness. Cut with an 8-inch bat-shaped cookie cutter. Place cookies 2 inches apart on lightly greased baking sheets.
Bake 15 to 17 minutes or until edges are lightly browned. Let cool on baking sheets 5 minutes; transfer to wire racks, and let cool completely. Pipe Royal Icing or prepared frosting onto cookies to outline; pipe two dots for eyes.
Note: To completely ice bat cookies, first pipe icing or frosting to outline. Add water to remaining icing, 1 teaspoon at a time, until mixture is the consistency of white glue. Pipe into center of cookies to fill in body. Let stand, uncovered, until completely dry.