Heat 3 cups (720 ml) of milk in a medium saucepan over medium-low heat until it is steaming, 5-6 minutes (do not allow it to come to a boil.)
While the milk is warming, whisk together the eggs, egg yolks, sugar, and a pinch of salt in a medium bowl until pale yellow in color, 1-2 minutes.
Once the milk has begun to steam, slowly pour half into the egg mixture while continuously whisking so the eggs don’t scramble. Once combined, slowly pour the egg and milk mixture back into the saucepan. 4. Keep warm over medium-low heat, stirring continuously for 15-20 minutes, until the mixture has thickened enough to coat the back of a spoon.
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Prepare an ice bath in a large bowl. Place a medium bowl in the ice bath and set a fine-mesh sieve on top. Pour the egg mixture through the sieve to remove any unwanted egg bits.
Remove the sieve and add the remaining cup of milk and heavy cream, and stir to combine. Let cool for 20 minutes, stirring every 5. Add the rum, if desired, then cover and refrigerate for at least 2 hours. The eggnog will keep in the fridge for up to 3 days.