Preheat oven to 325°F. Line baking sheets with parchment paper.
In a large bowl, cream butter and brown sugar until light and fluffy. Beat in egg yolks, eggnog and rum extract. Gradually mix in the flour. Chill for two hours.
Shape into 1.5 inch balls and bake until bottoms are golden brown, about 10 minutes - tops won't brown much. Cool completely.
For the Frosting: Beat In a large bowl, beat the first five frosting ingredients until blended; Mix in additional eggnog to reach desired consistency. Frost cookies and sprinkle with nutmeg if desired. Store in airtight containers.