Bring water to a boil in a saucepan. Add rice, butter, and salt; cook and stir until butter melts, then cover the saucepan and remove from the heat. Set aside until water is absorbed, about 5 minutes.
Stir milk, brown sugar, cornstarch, and vanilla into rice; bring to a simmer over medium heat while stirring. Cook until rice has the consistency of loose oatmeal, about 5 minutes. Ladle pudding into bowls and dust with cinnamon.