Pour cream and milk into a medium saucepan. Set over medium heat until warm and steaming. Slowly whisk about half of cream mixture into egg mixture, then whisk that mixture into cream mixture in saucepan. Set over medium heat.
Cook, stirring frequently, for 1 to 3 minutes until mixture coats back of a spoon or first bubble pops. Remove from heat and whisk in pumpkin purée, cinnamon and vanilla extract. Strain into a large bowl or container, then cool completely. Slowly whisk in rum. Refrigerate for about 1 hour until chilled.