In a medium sized heavy bottomed pot, combine milk, water, rice and salt. Bring to a boil on medium high heat, stirring frequently to avoid burning. When mixture comes to a boil, reduce heat and cook, covered, for 20-25 minutes or until rice feels mushy when squeezed between fingers.
In a separate bowl, whisk egg yolks together with sugar. Mixture will be thick. Spoon about half a cup of the hot rice mixture into the egg yolk/sugar mixture, bit by bit and beat well. Add back into the pot and continue cooking for another 1-2 minutes until pudding thickens slightly. Stir vigorously so the egg yolk mixture doesn't curdle. Remove from heat.
Add butter and vanilla and stir well. Serve warm or chilled with cinnamon sprinkled on top.