Preheat oven to 375°F (190°C). Line baking sheets with parchment paper.
In a medium bowl, whisk together flour, baking soda, salt, ginger, cinnamon, and cloves.
In a large bowl, cream together butter and brown sugar until light and fluffy. Beat in molasses and egg, then gradually add milk.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.
Drop by rounded tablespoons onto the prepared baking sheets.
Bake for 10-12 minutes, or until edges are lightly golden. Let cool on baking sheets for a few minutes before transferring to a wire rack to cool completely.
While cookies cool, prepare the frosting. In a medium bowl, beat the softened butter until smooth. Gradually add the powdered sugar, beating until combined. Stir in the eggnog and vanilla extract until smooth and creamy.
Once cookies are completely cooled, frost with the eggnog frosting.
Notes
Enjoy these Eggnog Frosted Ginger Molasses Cookies with a glass of milk or eggnog!