In large bowl, beat with a hand mixer or sturdy fork, the brown sugar, shortening and granulated sugar together until creamy. Add egg, water and vanilla; beat well.
In a medium bowl, combine oats, flour, salt, and baking soda, and mix well. Add dry ingredients to wet ingredients, and mix well.
Drop dough by rounded teaspoonfuls onto cookie sheets lined with parchment paper or Silpats.
Bake 12 to 15 minutes, or until edges are golden brown. Remove to wire rack. Cool completely.
Make icing:
Once cookies have cooled completely, make the icing. Place powdered sugar, 1 & ½ tablespoons eggnog and ¼ teaspoon eggnog extract in a medium bowl and mix well until all lumps have disappeared. Spoon or drizzle icing over the top of the cooled cookies.
Store tightly covered.
Notes
High altitude adjustment: Increase flour to 1&¼ cups and bake as directed.