Combine and heat gently: Put the rice, milk, sugar, and salt in a heavy-based pot. If using a vanilla pod, add it now. Heat over medium-low heat, bringing the milk barely to a simmer and stirring now and then. Do not let it boil hard.
Cook low and slow: Turn the heat low and cook for 35 to 45 minutes, stirring every few minutes and scraping the bottom of the pot. The rice will swell and the milk will thicken. The pudding is ready when the rice is completely soft and the pudding coats the back of a spoon. It thickens more as it cools, so take it off the heat while it is still a little looser than you want.
Finish with vanilla and butter: Remove the vanilla pod, if used, and stir in the vanilla extract and butter, if using, until melted. Taste and add a little more sugar if you want it sweeter.
Serve warm or chilled: Serve warm in bowls, dusted with cinnamon, or cool and chill for cold rice pudding. If it thickens too much on standing, stir in a splash of milk to loosen it. Some people like it with a spoon of jam, a few berries, or a drizzle of honey. All of them work.