Begin by creaming the softened butter and sugar in a large mixing bowl until the mixture becomes light and fluffy, which should take approximately 5 to 7 minutes. Next, incorporate the eggnog into the mixture. In a separate bowl, combine the flour, baking soda, and nutmeg. Gradually add this dry mixture to the creamed butter and sugar, stirring until well blended. Shape the dough into four rolls, each measuring 10 inches in length. Wrap each roll in waxed paper and refrigerate overnight.

When ready to bake, preheat your oven to 375°F (190°C). Unwrap the chilled dough and slice it into 1/4-inch thick pieces. Arrange the slices on ungreased baking sheets, ensuring they are placed one inch apart. Bake in the preheated oven until set, which will take about 8 to 10 minutes. Once done, remove the cookies from the oven and allow them to cool on wire racks.

To prepare the icing, beat the softened butter in a large bowl until fluffy. Gradually add the confectioners' sugar and eggnog, continuing to mix until the icing is smooth. Finally, frost the cooled cookies and store them in an airtight container to maintain freshness.