Gradually add sugar to eggs in a large glass mixing bowl, whisking until blended. Set aside.
Stir together half-and-half, milk, and salt in a Dutch oven over medium-low heat. Cook, stirring occasionally, 12 to 15 minutes or just until mixture begins to bubble around edges of pan. (Do not boil.)
Gradually stir half of hot milk mixture into egg mixture. Stir egg mixture into remaining hot milk mixture in Dutch oven.
Cook mixture over medium-low heat, stirring constantly, until mixture slightly thickens and a thermometer registers 160° (about 25 to 30 minutes). Remove from heat, and stir 1 minute. Pour mixture through a fine wire-mesh strainer into a serving bowl. Stir in bourbon, brandy, and vanilla; let cool 1 hour. Cover and chill at least 4 hours.
Beat whipping cream at high speed with an electric mixer until soft peaks form. Fold whipped cream into chilled eggnog, and sprinkle with nutmeg.
Eggnog With Coffee and Irish Cream: Prepare recipe as directed, substituting 4 1/2 cups chilled strongly brewed coffee and 3 cups Irish cream liqueur for bourbon and brandy. Prepare and fold in 2 cups whipped cream as directed. Omit nutmeg. Garnish with additional whipped cream and thin chocolate mints, if desired. Makes about 1 gal. Note: For testing purposes only, we used Baileys The Original Irish Cream and Andes Creme de Menthe Thins for chocolate mints.