Preheat oven to 350 degrees F.
In a large bowl, cream together butter and sugar until light and fluffy, approximately two minutes. Turn mixer to low and VERY slowly pour in oil. Take care to do this very slowly or you will end up with oil all over! Once combined, add in water and vanilla. Once combined add in eggs.
In a separate bowl combine dry ingredients. Slowly add the dry ingredients to the wet ingredients and mix until combined, but do not over mix.
Using a standard sized ice cream scoop, scoop dough onto a cookie sheet. Leave at least 3" between balls of dough. Using the back of a small cup (about 2" in diameter) dip the glass into sugar and than gently press into each ball of cookie dough, creating an indentation in the center. Repeat with remaining cookies.
Bake cookies for approximately 8 minutes. Remove from oven and if indentions have risen too much, quickly press into them with the back of a larger glass, very gently, immediately after removing them from the oven (they still need to be hot). Allow cookies to cool completely on the baking sheet before transferring.
In a large bowl combine powdered sugar, butter, International Delight Egg Nog, Hot Chocolate or Milk, along with any other items called for in the recipe, such as nutmeg, cocoa powder or peppermint oil. Stir until smooth (you don't need a mixer, but can use one). If frosting is too runny, add more powdered sugar, about 1 tablespoon at a time. If frosting is too thick, add more liquid, about 1 teaspoon at a time, until desired consistency is reached. Gently spoon icing onto indentation area of cooled cookies. If desired top with sprinkles, chocolate shavings, mini marshmallows, crushed candy canes or nutmeg. Allow frosting to set up for at least two hours (It will still be soft, this frosting does not dry hard to the touch).