Preheat oven to 350 degrees F (175 degrees C). Grease cookie sheets.
In a medium bowl, cream together the margarine and white sugar until smooth. Stir in the egg and 1 cup eggnog.
Combine the flour, baking powder, baking soda and salt; stir into the sugar mixture so it is well blended. Drop by rounded spoonfuls onto the prepared cookie sheet. Drop by rounded spoonfuls onto the prepared cookie sheets.
Bake for 8 to 10 minutes in the preheated oven. Allow cookies to cool on baking sheet for 5 minutes before removing to a wire rack to cool completely.
To prepare the icing, put the confectioners' sugar into a small bowl. Stir in the remaining eggnog one tablespoon at a time until the desired consistency is reached.
Spread onto cooled cookies and let dry before serving.
