In a large saucepan, combine the rice, milk, sugar, salt, cinnamon, and nutmeg. Stir well.
Bring the mixture to a boil over medium-high heat, stirring occasionally. Once boiling, reduce the heat to low, cover, and simmer for 30 minutes, stirring every 5-10 minutes to prevent sticking.
In a small bowl, whisk together the beaten eggs and vanilla extract. Gradually stir about 1 cup of the hot rice mixture into the egg mixture to temper the eggs. Then, pour the tempered egg mixture back into the saucepan with the rest of the rice mixture.
Continue to cook the pudding over low heat, stirring constantly, for another 10-15 minutes, or until thickened. Do not boil.
Remove from heat and serve warm, or chill in the refrigerator for at least 2 hours before serving cold. Garnish with extra cinnamon or nutmeg if desired.
Notes
This rice pudding is delicious served warm or cold. It can also be a base for fruit toppings.