In a small pot, add water and cinnamon sticks, bring to a boil and cook for 10 minutes. Set aside and discard cinnamon sticks. Place the rice and the cinnamon water in a large saucepan, over medium heat and cook for 5 minutes. Add salt, butter, vanilla extract, 2 cups of milk, and sugar. Stir well to mix and cook uncovered for approximately 15 minutes. Reduce heat to medium low. Add the remaining milk and condensed milk. Stir with a wooden spoon. Cook for 1 hour and 15 minutes, or until the rice pudding thickens to desire consistency.

Stir well and remove from heat, and let it cool at room temperature. The consistency should be very creamy. Refrigerate for a least 1 hour or overnight.