While rice is cooking, beat the egg in a bowl. Add the milk and stir well to mix. Add the egg mixture, vanilla extract and condensed milk to the rice and cook over low heat, stirring carefully, until most of the liquid evaporates and the rice reaches the desired consistency, about 30 minutes. Add the pumpkin puree and the pumpkin pie spice, stir until heated thoroughly remove from heat. Serve and add a dash of pumpkin spice, decorate with cinnamon sticks.