Combine the milk, sweetened condensed milk, flour, nutmeg, vanilla bean, and salt in the medium saucepan. Slowly bring to a boil over medium heat, stirring constantly with a whisk. Remove the piece of vanilla. Whisk the egg yolks in the small bowl, then gradually whisk one-third of hot milk mixture into egg yolks. Add yolk mixture to remaining hot milk mixture, stirring with a whisk. Cook over medium heat until slightly thickened, about 1 minute. Remove from heat and stir in the bourbon and brandy.

Serve hot in mugs, garnished with grated nutmeg. To serve cold, cover the surface of the eggnog with wax paper, and refrigerate at least 4 hours or overnight.