1.) In a medium heavy bottom saucepan, beat eggs and sugar until blended. Stir in 1 cup milk.
2.) Cook over low heat, stirring continuously, until mixture just coats the back of a spoon, about 15-20 minutes, or until temperature reaches 160°F, but do not allow to boil.
3.) Remove from heat, then stir in remaining milk and vanilla. Strain into a bowl, then refrigerate until completely chilled.
4.) Place chocolate syrup in a shallow plate. Dip rim of 6 martini glasses into chocolate syrup to coat rims. Refrigerate to chill.
5.) In a pitcher, stir together vodka, chocolate liqueur and eggnog.
6.) Serve in chilled martini glasses and top with grated nutmeg. Refrigerate leftovers up to 3 days.