Heat oil in a medium pan and saute eschallots for 3-4 minutes or until soft and translucent. Add pumpkin and saute for 4-5 minutes, stirring occasionally.
Stir in dry rice and saute for 3 minutes or until it begins to smell nutty. Add cayenne pepper and dried thyme.
Gradually add the hot chicken stock, a cup at a time. When the rice has soaked up the stock, add the next cup. Stir constantly and scrape up any rice that sticks to the bottom. It should take about 20-25 minutes to use up all the stock.
Remove pan from heat and stir in Parmesan and walnuts. Season with salt and pepper and serve immediately.