Rinse 1 cup rice under cold water until water runs clear to remove excess starch.
Place rinsed rice in a bowl and cover with 4 cups water; add the cinnamon stick and soak for 3–4 hours or overnight.
After soaking, transfer the rice, cinnamon stick, and soaking liquid to a blender.
Blend on high until the rice is finely ground and the mixture looks milky, about 1–2 minutes.
Strain the blended mixture through a fine-mesh sieve or cheesecloth into a pitcher, pressing solids to extract liquid.
Discard or reserve the rice solids for baking; return the strained liquid to the pitcher.
Stir in 1/2 cup granulated sugar and 1 tsp vanilla extract until dissolved.
Add 2 cups cold milk (or dairy-free alternative) and mix until smooth; adjust sweetness if needed.
Chill the horchata in the refrigerator for at least 1 hour to meld flavors.
Serve over ice and sprinkle with ground cinnamon or garnish with a cinnamon stick.