Put milk, 1⁄2 cup of the sugar, and salt into a medium heavy-bottomed saucepan. Scrape seeds from vanilla bean into pan, then add pod. Heat over medium heat, stirring until sugar has dissolved, about 10 minutes.
Whisk egg yolks in a mixing bowl until pale yellow. Slowly whisk 1 cup of the hot milk mixture into yolks. Gradually add egg–milk mixture back into milk mixture in saucepan and cook, stirring constantly with a wooden spoon, until thickened, about 5 minutes. Strain into a large bowl and set aside to let cool. Add bourbon and rum, cover, and refrigerate eggnog until cold.
Whisk egg whites in a mixing bowl until frothy, then gradually add remaining sugar, whisking constantly until stiff but not dry peaks form. In another bowl, whisk cream until stiff but not dry peaks form. Fold whites and cream into eggnog. Pour into cups and sprinkle with nutmeg.