In a 2-qt. saucepan, bring broth to a boil. Reduce heat to a gentle simmer.
Preheat a 3- or 4-qt. chef's pan over medium heat until rim of pan is hot to the touch. Add oil and heat for about 1 minute.
Add mushrooms and onion and stir often until vegetables soften and begin to stick to the bottom of the pan, about 15 minutes. If ingredients begin to scorch, add water, 1 Tbsp. at a time, as necessary.
Add the rice and sauté until the rice has begun to turn opaque, about 3 minutes longer.
Add wine to rice and continue to stir constantly; keep the liquid at a simmer, until most of the liquid is absorbed. Add thyme.
Ladle in 1/2 c. of simmering broth to rice and stir constantly until most of the liquid is absorbed. Continue cooking at a simmer and adding broth, about 1/2 c. at a time. Stir continuously with each addition. Allow all the liquid to be absorbed before adding the next. Continue this process until the rice is tender and creamy in appearance but still al dente, about 25 minutes in all.
Stir in the cheese and butter; season with salt and pepper to taste. Garnish with freshly chopped parsley and serve.