Step 1 In the bowl of a stand mixer, or in a large chilled bowl using a hand mixer, beat the cream to stiff peaks, about 3 minutes. Cover and refrigerate until needed.
Step 2 In a large bowl, whisk together the egg yolks and one-half cup of sugar until combined and frothy. Whisk in the rum.
Step 3Set the bowl over a large pot of simmering water, making sure the bottom of the bowl does not touch the water. Continue to whisk until the mixture increases in volume and thickens, 3 to 5 minutes. Remove the bowl from heat and place it over a large bowl of ice water, whisking to cool the mixture. Refrigerate until needed.
Step 4In the bowl of a stand mixer, or in a large bowl using a hand mixer, beat the egg whites until frothy. Gradually sprinkle over the cinnamon, allspice, nutmeg, ginger and the remaining 2 teaspoons sugar, a little at a time, and continue to beat until stiff peaks form. In a separate bowl, whisk together the milk and pumpkin purée until smooth. Set aside.
Step 5Fold the whipped cream gently into the chilled egg yolks, then, again very gently, fold in the beaten egg whites and milk mixture, each a little at a time, until incorporated. Refrigerate until ready to serve. To serve, pour into chilled glasses or eggnog cups (or demitasse cups), with a stick of cinnamon or pinch of nutmeg. The cocktail will keep for 1 day, covered and refrigerated.