Add the rice to a medium saucepan along with the water, cinnamon stick, and a pinch of salt. Place over medium-low heat and heat, stirring, until the water begins to simmer. Turn heat down to low and cook, stirring frequently, until the rice begins to soften and most of the water is absorbed, about 20 minutes.

Pour in the sweetened condensed and whole milk, then stir in the grated nutmeg. Continue to simmer on low heat, stirring constantly, until the rice is very tender and forms a thickened, pudding-like consistency. Turn heat off, remove the cinnamon stick, and stir in the vanilla extract.

Divide rice pudding between 4 8-ounce glasses, ramekins, or bowls. Press a piece of plastic wrap onto the top of each pudding to prevent a thick skin from forming then refrigerate until cooled and ready to serve, at least 1 hour. Right before serving, sprinkle each rice pudding with some of the ground cinnamon.