Preheat the oven to 350 degrees. In a large mixing bowl, combine the butter, sugar, and eggs with a hand mixer on medium speed for 2 minutes, or until well blended. Add the eggnog and rum, mix until combined.
In a small mixing bowl, combine the flour, baking powder, nutmeg, and salt. Gradually add the flour mixture to the egg mixture, mixing until well blended.
Transfer the biscotti dough to a floured board. Divide the dough in half, using more flour as needed to prevent sticking.
Place 1 dough ball onto the parchment lined cookie sheet and shape into a log approximately 14 inches long. Press down on each roll to flatten to approximately ½ inch tall. Bake for 25 minutes.
Remove the biscotti from the oven and cool for 15 minutes. Carefully cut the biscotti crosswise into slices approximately ½" in size getting approximately 36 slices. Place the cookie slices, cut side down, on the parchment lined cookie sheet. Bake for 10 minutes. Remove from the oven and turn cookies over. Return to the oven and bake for 10 minutes more, or until cookies are firm and lightly brown. Remove from oven and transfer the eggnog biscotti to a wire rack to cool completely.
In a small mixing bowl, whisk together the powdered sugar, rum, and homemade eggnog. Place a piece of parchment paper underneath the cooling rack and drizzle the cookies with the glaze forming lines over top of the biscotti. Store in a cool dry place up to 1 week or in the freezer for up to 3 months.