Stir milk, half & half, rice, sugar, and salt together in a cold pot. Place over heat and bring to a simmer and reduce heat down to medium-low to gently cook until rice is soft or cooked to your liking, about 30 minutes.
Remove from heat and whisk in cinnamon, nutmeg, vanilla, and butter. Crack an egg into a separate dish and whisk into hot rice pudding quickly to thicken without cooking.
Stir in heavy whipping cream and serve warm.
As rice pudding cools and is refrigerated, it will start to thicken. If you'd like to serve it cold, stir in some extra cream or half and half to thin to the consistency you'd prefer.