Preheat oven to 350℉ Line a large baking sheet with a silicone baking mat or parchment paper.
In a mixing bowl, combine butter, sugar, and eggs with a mixer for about until well blended about 1 minute. Mix in eggnog and rum just until blended.
In a separate bowl, combine flour, baking powder, nutmeg, and salt. Mix the flour mixture into the butter mixture gradually until all of the flour has been added and is well blended.
On the prepaid baking sheet shape each dough half into a long roll, approximately 14 inches long n prepared baking sheet spaced about 3 inches apart. With your hand, press down each roll so that they are about ½″ high.
Bake for 25 minutes or until golden brown. Remove loaves from pans and place on cooling racks. When biscotti is cool enough to handle, but still warm, using a serrated bread knife cut crosswise slices approximately ½ inch thick.
Place slices on the cut side back on the baking sheet. Bake for 10 minutes. Turn slices over and bake for another 10 minutes . Transfer to wire racks to cool completely.
Glaze
In a small bowl, mix together powdered sugar, rum and enough eggnog to reach drizzling consistency.
Place biscotti pieces about ½-1 inch apart on the baking sheet. Dip the tines of a fork into the glaze, allowing some of the excess to drip off. Hold the fork about 6 inches above the biscotti and sweep back and forth drizzling lines over all of the biscotti pieces. Allow to set. Store in an air tight container.