In a medium bowl, whisk the egg yolks and sugar together until the mixture is pale and thick.
In a saucepan, combine milk, cream, cinnamon sticks, and salt. Stir and heat until just below boiling, then remove from the heat and let sit for 5 minutes.
Slowly drizzle hot milk into the yolk mixture, whisking constantly, until all the milk is incorporated.
Return the mixture to the saucepan. Heat gently, stirring, until it reaches 160°F and thickens slightly.
Remove the cinnamon sticks. Strain into a bottle or container, cover, and refrigerate until cold.
Stir in vanilla extract and ground cinnamon before serving.
Notes
For an adult version, stir in a splash of rum, bourbon, or brandy before serving.
If the eggnog thickens too much in the fridge, thin it with a little extra milk.