Rinse rice 2-3 times with water until the water runs clear. Drain the water. Add the rinsed rice to a high-speed blender along with the almonds, cinnamon stick, and 1 cup of cold water.
Blend the ingredients to make a coarse mixture. The rice and almonds will just be crushed and that is what we want.
Transfer the mixture to a large glass bowl. Add the remaining 3 cups of water, pumpkin puree, and pumpkin pie spice, and mix well to combine.
Cover the mouth of the bowl using a lid or a plate and leave the bowl on the counter at room temperature for 8-10 hours.
Stir the mixture well and strain it through a fine-mesh strainer, a muslin cloth or a nut bag.
Discard the leftovers.
Stir in milk, vanilla extract, condensed milk, and sugar.
Chill the drink for a few hours before serving. Pour horchata into serving glasses, add a few ice cubes and sprinkle a pinch of pumpkin spice mix. Serve chilled.
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