Combine the milk, cinnamon sticks, cloves, and ½ teaspoon vanilla extract in a large saucepan. Place it over low heat for 5 minutes to infuse the flavors.
In a medium mixing bowl, whisk the egg yolks and sugar together for about 1 minute, or until the mixture becomes pale and slightly thickened.
Remove the cinnamon sticks from the milk and increase the heat to medium-high. Gradually whisk about ¼ of the hot milk into the egg yolk mixture to temper the eggs. Then, pour the egg yolk mixture back into the saucepan with the remaining milk, whisking continuously.
Cook the mixture over medium heat, whisking constantly, for 3 minutes, or until it thickens slightly. Do not let it boil.
Pour the hot eggnog into a large bowl or beverage container and allow it to cool at room temperature for 1 hour. If desired, return the cinnamon sticks to the mixture for additional flavor.
After cooling, stir in the heavy cream, 2 teaspoons vanilla extract, and nutmeg. Refrigerate the eggnog for at least 4 hours, or until fully chilled.
Video
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Notes
For an uncooked version, in a large mixing bowl, whisk the egg yolks, sugar, ground cloves, and nutmeg together. Add ½ teaspoon ground cinnamon in place of the cinnamon sticks. Gradually whisk in the milk, cream, and vanilla extract until thoroughly combined. Chill before serving.
Alcohol can be added before serving, to taste. Dark rum, cognac, and bourbon are all traditional options.