In a skillet set over medium-low heat, toast the rice and cinnamon stick, stirring occasionally, until they release a nutty aroma, about 5 minutes. Transfer to a blender and process into a fine powder. Add the sugar, vanilla extract, and 2 cups of water and blend well. Set a wet cheesecloth over a pitcher and strain in the rice mixture (discard the sediment), then stir in 2 more cups of water. Set in the fridge to chill completely.