1. In a large bowl beat the butter with an electric mixer on medium speed for 30 seconds. Add the granulated sugar and beat until light and fluffy, about 1 minute more. Beat in the egg and vanilla. Gradually beat in the flour.
2. Divide the dough in half. Wrap each half with plastic and chill for at least 1 hour, or up to 3 days.
3. Preheat oven to 375 degrees F. On a lightly floured surface roll out each dough portion to about 1/8-inch thickness. Using 3-inch hat or glove shaped cookie cutters, cut out 18 cookies, re-rolling scraps as needed. Place on an ungreased cookie sheet and bake for 8 to 10 minutes, or until lightly browned on the bottom.
4. Cool on the cookie sheet for 2 minutes, and then transfer to a wire rack; cool completely.
5. To decorate, In a medium bowl whisk together the powdered sugar and 3 tsp. milk, adding additional milk as needed to reach a thick but spreadable icing. Tint with a few drops of food coloring.
6. Add coconut for cozy cuffs and cute pom-poms, plus decorating sprinkles as desired.