Pour the rice and water into a blender. Let soak at room temperature 3 to 4 hours, or up to 12 hours, until rice is softened.
Strain and reserve the rice infused liquid and discard rice, or keep the rice in the water (see recipe notes).
Add in the evaporated milk, milk, sugar, vanilla extract, and ground cinnamon in with the rice and water.
Blend on high until completely smooth. Serve chilled or over ice.
