In a separate bowl, cream together shortening, vanilla, sugar and egg for about 2-3 minutes. Blend in dry ingredients and milk. Use cookie scoop to scoop onto prepared baking sheet.
Bake for 8 minutes. While baking, blend all of the icing ingredients together until smooth.
Remove cookies from oven, set 1 marshmallow half on each cookie and return to oven to bake for 1 more minute. Cool for 1-2 minutes, and then drop a spoonful of chocolate icing on each marshmallow topped cookie, letting it drizzle down the sides.
Let the cookies set up for a few minutes and then transfer to a wire rack to cool completely.