Beat the eggs well in a large glass bowl; set aside.
Whisk together cornstarch, salt, and sugar in large pan. Slowly stir in the milks.
Cook and stir over medium heat until the mixture comes to a boil. Continue cooking for 2-3 minutes or until thickened slightly.
Add the rice and bring the mixture back to a boil.
Slowly whisk about 3 cups of pudding into the beaten eggs. Pour mixture back into pan. Cook and stir till bubbly, then remove from heat and add vanilla.
If desired, stir in cinnamon and nutmeg.
Notes
If you like your rice pudding richer and saucier, use only 4 cups of rice. I usually serve it for breakfast, so I add 6 cups so it will be more filling.
The original recipe calls for 1 1/4 cups sugar. If you are serving this as a dessert, you could add the full amount.