In a small saucepan over medium heat, add the pumpkin puree, pumpkin spice, sugar, and water. Cook the mixture down slightly, stirring constantly, for approximately a few minutes. The mixture should still be a puree, so don’t cook it down too far!
Remove mixture from heat and strain through a fine-mesh sieve. Discard remaining puree. Cool completely before using in the refrigerator.
When you are ready to make the drinks, in a small bowl, stir together ingredients for the rimming sugar. Wet the rims of 4 champagne glasses and rim them with the pumpkin spice sugar. Set aside.
Add 1 1/2 oz of the strained pumpkin juice to the champagne glasses and give the mixture a stir.
Fill each glass to the top with champagne.
Top each with whipped cream, a cinnamon stick and a sprinkle of the remaining rimming sugar.