If using a cinnamon stick, place it in a small saucepan with 1 cup of water. Bring to a boil over medium-high heat, then remove from heat and allow it to steep for 10 minutes. This step may be omitted if using ground cinnamon.

Rinse the rice thoroughly under cold running water to remove excess surface starch. Drain completely before proceeding.

In a blender, combine the rinsed rice, steeped cinnamon water (including the cinnamon stick, if used), and the remaining 5 cups of water. Blend for approximately one minute, or until the rice grains have broken down and the liquid appears cloudy.

Pour the blended rice and cinnamon mixture into a large bowl or container. Cover and let it soak at room temperature for a minimum of 3 hours, stirring occasionally to ensure even flavor extraction.

Set a fine-mesh strainer lined with cheesecloth or a thin kitchen towel over a large bowl. Pour the soaked mixture through the strainer, pressing gently to extract as much liquid as possible. Discard the remaining rice solids.

Transfer the strained rice liquid to a pitcher. Add the sweetened condensed milk and vanilla extract. Whisk thoroughly until the milk is completely incorporated and the texture is smooth and uniform.

Refrigerate the horchata until well chilled, preferably for at least 1 hour. Serve cold over ice. Stir before serving to redistribute any natural settling.