Prep the Base: In a heavy-bottomed pot over high heat, combine the milk, heavy cream, sugar, rice, nutmeg, salt, and cinnamon stick. Bring to a simmer, stirring occasionally.
Simmer the Pudding: Once simmering, reduce the heat to very low. Simmer gently for 30-35 minutes. Stir every 4-5 minutes to prevent the rice from sticking.
Prepare the Liaison (Egg and Vanilla Mixture): In a small bowl, whisk together the eggs and vanilla. Set aside for now.
Continue Cooking: After 30 minutes, check the rice for doneness. A grain of rice should be soft with no crunch.
Temper the Eggs: Once the rice is cooked and the heat is turned off, slowly add some of the hot milk mixture to the liaison, one ladle at a time, whisking constantly to prevent curdling. Repeat until the liaison doubles in volume. Now pour the tempered egg mixture back into the pot all at once, whisking constantly.
Finish with Butter: Stir in the butter until fully melted and incorporated.
Serve warm or chill overnight: Transfer the pudding to a pan and cover with plastic wrap, poking a small hole to allow steam to escape. Refrigerate overnight.
