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Create the milk mixture. Make the egg mixture. Add the milk mixture slowly to the egg mixture and cook. Remove the eggnog from the burner. Add the remaining the ingredients. Cover the eggnog tightly with plastic wrap.
In a large saucepan, combine the milk, nutmeg, cinnamon, and vanilla. Cook on medium low heat until the mixture comes to a low boil, stirring occasionally. Failing to stir the milk mixture as it boils can cause it to burn and stick to the bottom of the pan, which may ruin the taste of your eggnog. In a large bowl, combine the egg yolks and the sugar, and beat or whisk until fluffy. If using an electric beater, beat the egg mixture on low speed for one or two minutes. Whisk the milk mixture into the room temperature egg mixture until the ingredients are well incorporated. Pour the combined mixture back into the saucepan. Cook it over medium heat until it becomes thick, which takes about three to five minutes. Eggs must be tempered, which means it's important to add the milk mixture gradually to the room temperature egg mixture. Combining them too quickly or adding the eggs directly to the hot saucepan can cause the eggs to curdle and ruin the eggnog. Let it cool at room temperature for about an hour or two, or until the mixture itself reaches room temperature Stir in the cream and rum or brandy. Mix the eggnog well. Feel free to skip the rum/brandy for a non-alcoholic version. However, keep in mind that non-alcoholic eggnog will spoil faster, so don't store it for too long before you drink it. Refrigerate it overnight to let the flavors combine, and serve with grated nutmeg and a cinnamon stick garnish. The eggnog is best if enjoyed within several days. Refrigerate unused portions and toss it after a week or if the eggnog develops a foul odor.