Bring water, salt, and butter to a boil in a medium saucepan. Add the rice, return to a boil, and then reduce the heat to the lowest setting. Shake the pan occasionally and cook until rice has absorbed the water but it still al dente, about 15 minutes.
Bring milk, sugar, vanilla, and a few dashes of cinnamon to a simmer in a separate saucepan. Add the cooked rice and cook at a simmer over medium-low heat until rice absorbs most of the milk and mixture starts to get thick and silky, about 10 to 15 minutes.
Transfer pudding to a large bowl and cool to room temperature. Place in refrigerator until cool and set. Serve with whipped cream and a dash more cinnamon.