Step 1

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.)

Step 2

Strain the mixture through a fine-mesh sieve into a large 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.

Step 3

Whisk the remaining 1/2 cup heavy cream in a large bowl until stiff peaks form. Slowly whisk the whipped cream into the chilled eggnog mixture, then whisk in the rum. Pour into glasses, sprinkle with nutmeg and serve with cinnamon sticks.