Put 1 cup uncooked rice, 6 cups water, and 2 cinnamon sticks into a blender. Blend on high until the rice starts to break up, about 1 minute (blend in two batches if your blender is small).

Pour the blended rice mixture into a bowl or jar and let it sit at room temperature for at least 3 hours.

Set a fine strainer over a large pitcher and pour the rice mixture through it to separate the liquid from the solids. Use the back of a spoon to press the solids in the strainer to extract as much liquid as possible. Discard the rice solids and cinnamon sticks.

Add 1 cup whole milk, 1 tablespoon vanilla extract, and 2/3 cup sugar to the strained liquid. Stir until the sugar is fully dissolved.

Chill the horchata in the refrigerator before serving.