Beat egg yolks in a large bowl with an electric mixer until smooth. Beat in sugar until dissolved. Mix in bourbon, milk, heavy cream, and cognac. Add peach brandy, apricot brandy, and light rum; mix well.
Beat egg whites with an electric mixer in a separate bowl until stiff peaks form. Fold into the egg yolk mixture. Fold in dark rum. Sprinkle nutmeg on top and chill until serving.