In a large pot, bring the water to a boil and add the rice and salt.
Reduce heat to low, cover the pot and simmer for about 20 minutes until the rice has absorbed most of the water.
Add the milk and cinnamon stick to the pot and stir well.
Simmer the mixture over low heat, stirring occasionally to prevent sticking, for about 45 minutes to an hour, or until the rice is soft and the pudding is thick and creamy.
Remove the cinnamon stick and stir in the butter, sugar, and vanilla extract.
Serve the Riisipuuro warm or chilled, garnished with a sprinkle of cinnamon, if desired.