Bring water, rice, and salt to a boil in a saucepan over medium-high heat. Reduce heat to low, cover, and simmer until tender, about 20 minutes. Remove from heat; pour in milk and sprinkle in sugar. Stir with a whisk until the thin layer of cooked-on starch at the bottom of the pan is cleared and incorporated into the mixture, 2 or 3 minutes.

Place pan over medium heat, stirring frequently, until pudding reaches your desired level of doneness and creaminess, 8 to 10 minutes. The longer it cooks, the thicker and stickier it will be. Remove from heat. Stir in vanilla and cinnamon. Very quickly whisk in egg yolk (to prevent it from cooking). Whisk for about 1 more minute. Add dried cherries and butter; stir thoroughly.