Put rice, milk, sugar and salt in a heavy medium saucepan. Bring slowly to a gentle simmer over medium heat, stirring to keep rice from sticking.

Lower heat and cook uncovered for 30–40 minutes, stirring frequently and scraping the base and corners. The rice should be tender and the milk creamy. Do not let the milk boil vigorously or form a scorched layer. Add a small splash of milk if thickening before the rice is tender.

Remove from heat and stir in vanilla. Let stand for 15 minutes, stirring once or twice. The pudding thickens as it cools; stop cooking while it is still a little looser than the texture you want.

Divide into six small bowls. Serve warm with optional cinnamon, or cool promptly in shallow containers and refrigerate. Stir in a little milk to loosen a chilled portion if needed.