Melt butter and heat the oil in a Dutch oven set over medium heat.
Stir in the onions and cook for 3 minutes, stirring often until onion is soft.
Stir in the garlic and cook for 15 seconds, or until fragrant.
Add the mushrooms and season with salt and pepper; continue to cook for 5 more minutes, or until mushrooms are caramelized.
Add the thyme; stir and cook for 1 more minute.
Spoon out about ⅓ of the mushrooms and set them aside for serving.
Remove the pot from heat and transfer the ingredients to a high powered blender or food processor.
Add 2 cups of broth to the blender and process the mixture until smooth.
Return mixture to the Dutch oven; set the Dutch oven over high heat and stir in remaining broth. Bring to a boil.
Stir in the rice; reduce heat to medium-low and cover the pot.
Continue to cook for 25 to 30 minutes, or until rice is tender.
Stir in the milk and heavy cream; then, stir in the parmesan cheese.
Cook for about 6 to 8 more minutes.
Taste for salt and pepper and adjust accordingly.
Ladle into bowls, top with reserved mushrooms, and serve.