Set a medium metal bowl in a large bowl of ice water and have a fine sieve at the ready.
Combine the milk, half the sugar, and the nutmeg in a 3-quart saucepan. Set the pan over medium heat, stirring occasionally to encourage the sugar to dissolve. Heat the mixture through but do not allow it to boil (this is key to avoid making eggnog scrambled eggs!). Remove from the heat.
Put the egg yolks and salt in a small heatproof bowl and gently whisk to break up the yolks. Whisk in the remaining half of sugar and beat the mixture until the yolks become very pale and light. Gradually whisk in the warm sweet milk mixture. Pour the yolk mixture back into the saucepan, and whisk to combine.
Cook over medium-low heat (slow and steady), stirring constantly with a clean wooden spoon until the custard thickens enough to coat the back of the spoon and hold a line drawn through it with a finger, about 8 minutes. An instant-read thermometer should register 170°F–175°F. Do not let the sauce overheat or boil because then you will probably curdle the mixture, making eggnog scrambled eggs. Immediately strain the sauce through the sieve into the bowl set in the ice-water bath.
Gently whisk in the rum and vanilla extract. Stir the drink occasionally until cool, about 10 minutes. Transfer it to a large container. Cover and refrigerate for about 1 hour, until the eggnog is velvety and slightly thick.