Prepare the rice mixture: Rinse the uncooked, raw rice in cold water and drain it.
Soak the ingredients: In a bowl or container, combine the raw rinsed rice, roasted sesame seeds, roasted pumpkin seeds, and cinnamon stick. Add cold water, cover, and refrigerate the mixture for at least 7 hours or overnight. Blend: Transfer the rice, seed, cinnamon and water mixture to a blender. Blend on high.
Strain the mixture: After blending, strain the water from the rice and seed mixture using a fine mesh sieve or cheesecloth.
Strain again: For the smoothest texture, strain rice mixture one more time using a metal sieve or cheesecloth. Squeeze well if using cheesecloth to remove all the liquid. Discard solids.
Add remaining ingredients: Add milk, pumpkin puree, vanilla, sugar, pumpkin spice, and a pinch of salt to strained liquid.
Adjust the consistency: Pour the horchata into a pitcher, adding water until you reach your desired consistency (typically there should be about a quart in total). Stir well and refrigerate until cool.
Serve: Stir and serve horchata over ice in tall glasses.
Garnish: Garnish with cinnamon sticks and a dusting of ground cinnamon.Or, dip the rim of the glass in honey or syrup before filling with horchata. Then dip the rim in crushed graham crumbs or toasted sesame seeds.