In a medium mixing bowl, whisk the eggs until they are light and frothy.
Gradually add the sugar to the eggs while whisking, until fully combined and slightly thickened.
In a saucepan over medium heat, combine the milk and heavy cream. Heat gently until warm but not boiling.
Slowly pour about half of the warm milk mixture into the egg and sugar mixture while whisking continuously to temper the eggs.
Once combined, return the egg mixture to the saucepan with the remaining milk. Cook over low heat, stirring constantly, until the mixture thickens slightly and coats the back of a spoon.
Remove from heat and stir in the vanilla extract and ground nutmeg.
If using alcohol, stir in the bourbon or rum at this stage.
Strain the eggnog through a fine mesh strainer into a pitcher or large jar to remove any cooked egg bits.
Let the eggnog cool to room temperature, then refrigerate until chilled, about 1-2 hours. Serve cold with a sprinkle of additional nutmeg if desired.