Place the 1 1/2 cups of uncooked long-grain white rice, 2 cinnamon sticks, and 2 cups of the hot water in a blender.
Blend for approximately 1 to 2 minutes, or until the rice and cinnamon sticks have been coarsely ground. This initial blending releases the cinnamon flavor while beginning to break down the rice.
Pour the remaining 2 cups of hot water into the blender.
Blend the mixture again to further reduce the rice and cinnamon particles and distribute their flavors throughout the liquid.
Transfer the blended rice-and-cinnamon mixture to a pitcher or container fitted with a lid.
Place it in the refrigerator and allow it to soak overnight, or for a minimum of 8 hours. The extended resting period gives the cinnamon time to infuse the liquid while the rice contributes a fuller, creamier consistency.
Once the soaking period is complete, strain the mixture through a fine-mesh strainer or cheesecloth into a clean pitcher or large bowl.
Discard the remaining rice and cinnamon solids so the resulting horchata base is as smooth as possible.
Pour 2 cups of milk or unsweetened almond milk into the strained rice liquid. Add 2 teaspoons of vanilla extract, 2 teaspoons of ground cinnamon, and 1/3 to 1/2 cup of granulated sugar.
Stir thoroughly until the ingredients are evenly incorporated and the sugar has dissolved. Taste and adjust the amount of sugar within the stated range as desired.
Refrigerate the finished horchata until thoroughly chilled. Stir well immediately before serving, then pour over ice.