In a medium sized pot add the rice, cinnamon stick, and water. Place over medium heat and bring to a boil, stirring occasionally.
Once boiling cover with a lid, reduce heat to low, and simmer for 10-12 minutes. At this point most of the water should be absorbed.
Stir in the milk, pumpkin puree, and pumpkin pie spice. Turn heat to medium-low, cover, and continue cooking for 10 minutes.
Remove the lid and stir in the vanilla and brown sugar. Continue cooking (can raise heat to medium) until the rice pudding has reached a thick consistency, about 20 mins or so. While the rice pudding is thickening, make sure to stir it occassionally to prevent it from sticking to the bottom of the pot.
Once thick, remove from heat. Discard the cinnamon sticks. Serve warm or cold, topped with a sprinkling of chopped pecans.