Place the rice in a blender or spice grinder and process until thoroughly pulverized. Take your time doing this - this is a critical step and the more finely ground the rice, the better the horchata will be. Transfer to a large bowl or pitcher and add the almonds, cinnamon, and lime zest. Add about 3 cups of hot water, cover, and let stand overnight.
The next day, transfer the mixture to a blender and blend until mixture is as smooth as it will get, again taking your time to allow to get as smooth as possible. Add 2 cups additional cool water and blend again until thoroughly combined. Pour mixture through a strainer lined with 3 layers of fine cheesecloth set over a bowl, little by little, pressing on solids and stirring to help mixture pass through the cloth. The more slowly you strain the mixture, the smoother the horchata will be. Transfer to a pitcher and add 2 cups more water, until the mixture is a pleasant, smooth consistency. Add sugar and vanilla extract, to taste. Cover and refrigerate until thoroughly chilled.