In a heavy saucepan, combine eggs, granulated sugar and salt; gradually whisk in milk. Stirring constantly, cook for 10 to 15 minutes over medium heat, until mixture just coats a metal spoon. Remove from heat, and stir in 1 teaspoon vanilla extract. Place saucepan in a tin of ice water to let mixture cool, then transfer to the fridge to chill completely (this can be done a day ahead of time).
In a chilled bowl, combine cream, icing sugar and 1/2 teaspoon vanilla extract. Beat until stiff peaks form. Stir in rum, if using. Fold whipped cream mixture into the cooled egg mixture. Pour into serving bowl, and sprinkle with nutmeg. Serve immediately.
If you don't add rum, you may want to add some milk to the eggnog to thin it out a bit.