Add rice and water to the pitcher of a blender and blend until the rice is chopped into small pieces. About 1-2 minutes.
Pop cinnamon stick into the pitcher, cover, and refrigerate for a minimum of 4 hours, but overnight is best.
Once mixture has soaked, strain the rice milk through a fine mesh sieve lined with cheesecloth that's been placed over a pitcher.
While milk is straining, whisk together pumpkin puree, coconut milk and sugar in a small sauce pan and cook over medium heat until the sugar dissolves. About 3-4 minutes.
Remove pumpkin mixture from heat and stir in the vanilla extract.
Stir the pumpkin mixture into strained rice milk until well mixed.
To serve, add ice and 1 ½ ounces of rum to a glass rimmed in cinnamon sugar, top off with with the pumpkin spice horchata, and stir with a cinnamon stick.