Thinly slice one container of mushrooms; chop the remaining container of mushrooms. Keep them separate, and set aside.
In a medium saucepan, bring stock and thyme to a simmer over medium-high heat; reduce heat to low; keep warm.
In a Dutch oven, heat 2 tablespoons oil and 1 tablespoon butter over medium heat until butter melts.
Add sliced mushrooms to Dutch oven. Sprinkle with ¼ teaspoon salt, ¼ teaspoon pepper, and red pepper flakes.
Cook, stirring occasionally, until mushrooms are golden brown, about 5 minutes. Remove from pan, and place on a plate. Set aside.
Add remaining ¼ cup oil and 2 tablespoons butter to Dutch oven. Add chopped mushrooms, onion, and remaining 1 teaspoon salt and ¼ teaspoon pepper.
Cook, stirring occasionally, until onion and mushrooms are tender, about 5 minutes.
Add garlic and rice; cook, stirring occasionally, until rice is toasted, about 2 minutes.
Add wine to pan; scrape to release any browned bits. Simmer until wine is evaporated.
Reduce heat to medium low. Add about 1 cup of warm stock to pan. Cook, stirring constantly, until broth is mostly evaporated.
Continue adding broth, about 1 cup at a time, stirring until evaporated before between additions, until rice is tender, about 25 minutes.
Remove from heat; stir in Parmesan cheese.
Top servings with cooked sliced mushrooms; garnish with additional Parmesan.