Rinse the rice in a colander under cold water. Transfer the rice to a bowl and add the cinnamon sticks and 4 cups of water. Cover and refrigerate overnight, or for at least 4 hours.
After soaking, remove the cinnamon sticks (a few small pieces can remain). Blend the rice and soaking liquid in two batches, using half of the mixture each time and including some cinnamon. Blend for about 4 minutes, until very smooth and watery.
Strain the blended mixture through a very fine strainer or cheesecloth into a pitcher, pressing the solids with a spoon or spatula to extract as much liquid as possible.
Stir in the canned milks, vanilla, allspice, and the remaining 4 cups of water. Mix well until fully combined.
Taste and adjust with sugar or additional water as needed.
Chill, stir well, and serve over ice. Enjoy!
Notes
I was inspired by MuyDelish on her Traditional Mexican Horchata. my wife personally like a little bit of allspice to this because it ties in the cinnamon and vanilla flavors together.