In a medium saucepan, combine the rice, milk, and salt. Bring to a boil over medium-high heat.
Reduce heat to low, cover, and simmer for 30 minutes, stirring occasionally, until the rice is tender and the mixture has thickened.
In a small bowl, whisk together the sugar and beaten eggs.
Gradually whisk 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 remaining rice mixture.
Cook over low heat, stirring constantly, for 5-7 minutes more, until the pudding is thick and creamy. Do not boil.
Stir in the vanilla extract. Serve warm, or chill in the refrigerator for at least 2 hours before serving. Sprinkle with cinnamon if desired.
Notes
This rice pudding is delicious served warm on a chilly evening or cold on a warm day. It can be garnished with a sprinkle of cinnamon or nutmeg.