Rinse the long-grain white rice thoroughly. In a large bowl, combine the rinsed rice, the 3-inch cinnamon stick, and 4 cups of water. Cover and let it soak at room temperature for at least 4 hours, or preferably overnight.

Transfer the soaked rice, cinnamon stick, and soaking water into a high-speed blender. Blend on high until the mixture is as smooth as possible, about 2-3 minutes. The liquid will be milky white.

Pour the blended mixture through a fine-mesh sieve lined with cheesecloth into a large pitcher or bowl. Press down on the solids with a spoon or spatula to extract as much liquid as possible. Discard the remaining rice and cinnamon pulp.

Add the 2 cups of fresh water, granulated sugar, and vanilla extract to the strained liquid. Stir well until the sugar is completely dissolved.

Cover the pitcher and refrigerate for at least 2 hours to chill thoroughly. Serve the Mexican Horchata over ice, optionally garnished with a pinch of ground cinnamon.