In a large mixing bowl, beat the eggs with the granulated sugar until the mixture is pale and slightly thickened.
In a saucepan over medium heat, combine the milk, heavy cream, and spices (cinnamon, nutmeg, ground ginger). Stir occasionally until the mixture is warm but not boiling.
Take the warm milk mixture off the heat. Slowly pour it into the egg and sugar mixture while whisking continuously to temper the eggs.
Once combined, return the mixture to the saucepan. Cook over low heat, stirring constantly until it lightly thickens and reaches around 160°F (71°C). Do not let it boil.
Remove the mixture from the heat, add the vanilla extract and cinnamon syrup, and stir well. Allow the eggnog to cool to room temperature.
Once cooled, transfer the eggnog to a serving pitcher and chill in the refrigerator for at least 2 hours (or overnight for the best flavor).
Serve cold in festive glasses, topped with whipped cream and a dusting of cinnamon and nutmeg.