Beat egg whites in a glass, metal, or ceramic bowl until foamy. Add 1 cup sugar gradually, continuing to beat until stiff peaks form. Lift your beater or whisk straight up: the egg whites will form sharp peaks.
Beat egg yolks and remaining 1 cup sugar in a separate bowl until frothy and lightened in color.
Fold egg whites into egg yolk mixture gently; pour into a large punch bowl or pitcher. Stir in rum, half-and-half, and milk.