Combine all ingredients in the bowl of a stand mixer fitted with the paddle attachment (or in a large mixing bowl if using a hand mixer). Mix on low, scraping the bowl occasionally, until the mixture forms a cookie dough. Be sure to start on low or the dry ingredients will fly everywhere!

Divide the dough in half. Place a large sheet of waxed paper on a large cutting board or flat surface. Place the cookie dough on the wax paper. Cover with another large sheet of waxed paper. Press into a flat disk, then roll out the dough between the papers to about 1/4 inch thickness. Chill in refrigerator for at least 30 minutes.

Roll out the second half of the dough the same way and chill as well.

Preheat oven to 375°F. Cut out cookies into shamrock shapes with a cookie cutter. Place on parchment-lined baking sheet.

Bake for 6-10 minutes or until the bottoms are lightly browned. Cool completely before frosting.

Prepare frosting by beating butter until smooth, then gradually adding powdered sugar, vanilla, and milk until desired consistency. Tint with green coloring.

Frost cookies and dip in green sanding sugar. Allow to set.