In a medium pot over medium-high heat, bring the broth/stock to a simmer.
In a large, heavy bottomed pot over medium heat, add the olive oil and butter over medium heat. Once the butter is melted, add the onion, garlic, and thyme, stirring occasionally. Cook until the onions are soft and translucent.
Add the rice, stirring to coat each grain with the butter and oil. Stirring fairly constantly, wait until you smell a nutty aroma. That lets you know that the rice has been toasted a bit.
A couple of ladles at a time, add the warmed broth/stock to the rice. Stir constantly. Once the liquid has been absorbed, add another couple of ladles. More stirring. Repeat this until most or all of the liquid had been incorporated and absorbed, and the rice has a nice tender bite to it.
Add the pumpkin puree, stirring to incorporate well. Add the spices and cheese. Stir to combine. Add salt and pepper to taste.