Rinse the rice under running water until the water runs clear.
Add the rinsed rice, almonds, cinnamon stick, and 3 cups of water to a large bowl or pitcher. Cover and let it sit in the refrigerator for at least 8 hours, up to 24 hours.
Pour the bowl of ingredients into a large blender and add in the 3 cups of remaining water. Blend on high until completely smooth, about 2 to 5 minutes depending on the power of your blender.
Strain the blended mixture through a fine mesh strainer or nut milk bag to remove any solids. This may require straining 2 to 4 times to remove all the grittiness.
When completely smooth, pour it into a large pitcher and mix in the milk, sugar, and vanilla extract. Stir until the sugar has fully dissolved. Serve over ice.
Notes
Ceylon cinnamon: Also referred to as "Mexican cinnamon." Ceylon cinnamon is light brown, fragile, and has a lot of layers that are easily broken, as opposed to Cassia cinnamon which is very hard and is not easily broken in half. Ceylon cinnamon can be easily found in Hispanic grocery stores or online.
Separation of the water and the rice mixture is completely normal, especially if it's been sitting in the fridge for a day or two. Just mix it together with a large spoon and serve as you normally would.
To make it dairy free, use a dairy milk alternative like almond milk, soy milk, coconut milk, oat milk, or cashew milk.
To make it nut free, you can simply omit the almonds. No need to replace them.
If you don't want to use granulated sugar, you can use a different sweetener like honey, agave, or even sweetened condensed milk to make it extra creamy.