1. Prepare cookies: In a mixing bowl, cream butter and sugar with electric mixer until light and fluffy. Add egg and vanilla extract; mix well. Combine remaining cookie ingredients in a small bowl, then slowly add to the butter mixture, mixing until blended. Gather dough into a ball, cover with plastic wrap and place in refrigerator to chill for at least 1 hour, or up to a day.
2. Preheat oven to 375ºF. Grease a baking sheet. Remove dough from refrigerator and let sit at room temperature for a few minutes, until softened enough to roll out. Lightly dust a work surface with flour and sifted confectioners’ sugar. Roll out the dough to 1/8-inch thickness; cut out cookies with a shamrock cookie cutter. Collect trimmings and repeat. Arrange cookies on baking sheet, 1 inch apart, and bake for 8 to 10 minutes, or until light brown. Cool cookies completely on wire racks before decorating.
3. Prepare icing: Mix green food coloring with milk. Combine with confectioners’ sugar, stirring until color is evenly distributed and icing is smooth. Using a spoon or a small brush, frost cookies with green icing. If desired, place icing in a zip-top plastic bag, snip off a corner with scissors and pipe decorations onto cookies.
4. To apply cookie to stick, pipe or spread icing on the back of the cookie and adhere the stick by sandwiching two cookies together. Decorate after stick is in place and serve!