Rinse under cold water to remove all starch until water runs clear. Add rice, water, butter, salt and one cinnamon stick to a medium-sized pot.
Cook rice on medium-high heat. Bring the rice mixture to a boils. Next, reduce heat to medium-low and reduce to a simmer
Cook the rice for about 10 minutes. The liquid should all be absorbed.
In a medium mixing bowl, whisk together the milk and eggs. Add the sugar, vanilla and salt and whisk.
Add the milk mixture to the rice. Stir frequently to avoid the eggs setting. Bring pudding to a simmer then cook for an additional 30 minutes. The pudding will be thick and creamy.