In a saucepan, soak the rice and cinnamon stick in the 4 cups of water for one hour.
After soaking the rice and cinnamon, bring the rice mixture to a boil, uncovered.
When the mixture begins to boil, lower the heat to medium and cook for 12 minutes (by this time, the water should be nearly evaporated).
While the rice is cooking, beat the egg in a bowl and add the milk, stirring to mix.
Add the egg mixture, vanilla extract and condensed milk to the rice mixture and cook over low heat while carefully stirring until most of the liquid evaporates and the rice reaches the desired consistency; this takes about 30 minutes.
Add the pumpkin puree and the pumpkin pie spice.
Stir until thoroughly heated and remove from heat.