In a blender, combine the rice and warm water. Blend for 1 to 2 minutes until the rice breaks up but does not form a powder. Let the rice mixture stand in the refrigerator for 6 hours or up to overnight.
Strain the rice mixture into a container through a fine-mesh sieve or several layers of cheesecloth to remove the rice solids.
Pour the rice water into a blender and add the milk, vanilla and almond extracts, cinnamon, and sugar. Blend for 1 minute until well combined and smooth. Serve the horchata over ice with cinnamon sticks as stirrers.