Grind rice into very fine, cornmeal-sized pieces using a blender or coffee grinder.

Transfer ground rice to a medium bowl, pour warm water on top, and drop in cinnamon sticks. Cover and refrigerate, 8 hours to overnight.

Remove the cinnamon sticks. Place mixture back in the blender and puree until as smooth as possible.

Line a sieve with a couple layers of cheesecloth and strain the mixture. Squeeze cloths to extract as much flavor as possible.

Stir in milk, sugar, and vanilla extract until sugar has dissolved. Chill completely, at least 2 hours.

Serve over ice, garnished with cinnamon.