1/4 teaspoon freshly grated nutmeg, plus more for topping
1/2 cup cold rum or brandy (optional)
Cinnamon sticks, for serving
Whisk the eggs, egg yolk and sugar in a bowl until combined. Heat the milk and 1/2 cup heavy cream in a medium saucepan over medium heat until steaming; do not boil. Remove from the heat. Whisk 1 cup of the hot milk mixture into the egg mixture, then whisk the egg mixture back into the saucepan. Cook over medium heat, stirring constantly with a wooden spoon, until the mixture is thick enough to coat the back of the spoon and a thermometer registers 165 degrees F, about 5 minutes. (Do not boil or the eggs will curdle.)
Strain the mixture through a fine-mesh sieve into a medium bowl and stir in the vanilla and nutmeg. Let cool at room temperature, about 1 hour. Cover and refrigerate until cold, at least 2 hours or up to 1 week.
Whisk the remaining 1/2 cup heavy cream in a large bowl until stiff peaks form. Slowly whisk in the chilled eggnog mixture, then whisk in the rum. Pour into glasses, sprinkle with nutmeg and serve with cinnamon sticks.