In a blender, grind the rice until it has the consistency of very coarse polenta. Pour into a bowl, cover with water, add the cinnamon stick, cover and refrigerate for 8 hours or overnight.
Remove the cinnamon stick, pour the mixture back into the blender and puree the rice and water really well. Strain the mixture through a sieve lined with cheesecloth, pressing it to extract as much of the rice flavour as possible.
Stir in the sugar and milk, mixing until the sugar is dissolved. Taste, and adjust sweetness, if necessary. Refrigerate until completely chilled. Serve over ice, topped with freshly grated nutmeg or cinnamon.