Step 1: Rinse the rice thoroughly under cold water until the water runs clear. Combine the rinsed rice and broken cinnamon sticks in a large bowl. Pour 4 cups of hot water over the rice and cinnamon, ensuring everything is submerged.
Step 2: Cover the bowl and let the mixture soak at room temperature for at least 4 hours, or preferably overnight (8 hours). Soaking allows the rice to soften and the cinnamon flavor to infuse.
Step 3: Transfer the soaked rice, cinnamon sticks, and the soaking water into a high-speed blender. Add the vanilla extract. Blend on high for 3-5 minutes, until the mixture is very smooth and the rice grains are completely pulverized into a fine liquid.
Step 4: Place a fine-mesh sieve over a large pitcher or bowl. Line the sieve with a cheesecloth or a clean kitchen towel (ensure it's lint-free). Slowly pour the blended horchata mixture through the cheesecloth into the pitcher. Gather the edges of the cheesecloth and squeeze out as much liquid as possible from the rice pulp. Discard the remaining pulp.
Step 5: Stir in the sweetened condensed milk and evaporated milk into the strained horchata liquid until thoroughly combined. Taste and adjust sweetness if desired, adding a little extra sugar if preferred.
Step 6: Chill the horchata for at least 2-3 hours before serving to allow the flavors to meld and for it to become perfectly refreshing. Serve over ice, garnished with a sprinkle of ground cinnamon. Store any leftovers in an airtight container in the refrigerator for up to 3-4 days.