Mix powdered sugar, butter, vanilla and egg in a large mixing bowl. Stir in remaining ingredients.
Wrap tightly in saran wrap and place in fridge for 2-24 hours to chill.
Once chilled, remove from fridge and roll out into a ½ inch thick on a lightly floured surface (I use powdered sugar instead of flour)
Cut into a circle shape with a 2½ inch circle.
Roll out slightly larger to accommodate the marshmallow.
Bake at 375 degrees for about 7 minutes.
Cool on a rack.
For the icing:
Mix all ingredients and test consistently. Add slightly more milk for a thinner texture or more powdered sugar for a runny icing. The consistency should be thick enough to coat a spoon but thin enough to spread on a cookie.
Recipe by My Frugal Adventures at https://myfrugaladventures.com/2013/12/melting-snowman-sugar-cookies/