1 cup white rice, thoroughly rinsed
1/2 cup raw or blanched almonds, or 1/4 cup almond meal
1/4 to 1/2 tsp ground cinnamon
4 to 8 whole green cardamom pods
Scrapings of 1 vanilla bean or 1/4 tsp vanilla extract
3 cups boiling water
Using blender, grind rice and cardamom pods to powder. If using whole almonds, grind them to meal. Combine all ingredients in blender and blend until very smooth (at least 3 minutes on high speed--seriously, turn it on and walk away for a bit). Transfer to container with airtight lid. Seal and let sit in fridge at least 6 hours and ideally 12 to 24 hours.
Most of the sediment will have settled to the bottom of the container; leave it there. Strain liquid into a 2-quart jar through a coffee filter, several layers of damp cheesecloth, a cheesecloth-lined strainer, or a very fine strainer to get all the grit out. (It will be necessary to strain several times, so be prepared to wash out the cheesecloth or strainer, or use several coffee filters in succession. Note to self: The small wire mesh strainer is ideal for this; keep a bowl of hot water handy for rinsing it so you don't have to run back and forth to the sink.) The liquid will still be slightly chalky no matter what you do, but the final dilution will help.
Boil 1/2 cup water and 1/2 cup sugar (or more, if you want leftovers) to make a syrup.
To the horchata base, add:
3 cups cold water (or to desired consistency)
1/4 cup sugar syrup (or to taste)
Serve over ice. Store in fridge for up to five days. Sediment will settle during storage; shake well before serving to improve flavor and give a creamier (albeit also slightly chalky) texture.
Thoughts on modifications:
1) It might not be unreasonable to just blend everything without pre-grinding the rice, cardamom, and almonds. I've also seen variations that call for using cooked rice, or for soaking the rice and almonds whole and then blending them up on day two; I'll probably try both versions at some point. For the latter variation, I could also try toasting the rice and almonds, which is highly inauthentic but flavor-enhancing.
2) This version takes advantage of the sediment settling out overnight, rather than blending it back into the liquid and then straining it out again. If I try a version that calls for blending on day two, I will definitely let it sit for at least half an hour after blending to reduce the annoyance factor of the straining step.
3) It uses ground cinnamon rather than a cinnamon stick for more even distribution of flavor (and also because cinnamon sticks are almost always too dried up to be of much use in a cold recipe). However, the ground cinnamon texture is not entirely delightful. Another option would be to put a reasonably fresh cinnamon stick in the first batch of water and let it boil for a while to extract the flavor. You could also put one in the hot sugar syrup and leave it to steep in there for half an hour or an hour while the syrup cools; after making the horchata, you'll have leftover cinnamon-infused syrup, which is never a bad thing, but it does mean that increasing the sweetness means increasing the cinnamon and vice versa.