Heat 2 tablespoons olive oil in a large skillet over medium-high heat. Add mushrooms in a single layer and cook undisturbed for 3-4 minutes until golden brown. Stir and cook another 2-3 minutes. Season with salt and pepper, then remove and set aside.
In the same skillet, add remaining olive oil and 2 tablespoons butter. Sauté onion until softened, about 5 minutes. Add garlic and cook for 1 minute until fragrant.
Add cauliflower rice and thyme to the skillet, stirring to coat. Pour in white wine and cook until mostly absorbed, about 2 minutes.
Begin adding warm chicken broth one ladle at a time, stirring frequently and allowing each addition to be mostly absorbed before adding more. This process should take about 15 minutes total.
When cauliflower is tender and mixture is creamy, stir in the cooked mushrooms, remaining 2 tablespoons butter, and Parmesan cheese. Stir until cheese is melted and everything is well combined.
Taste and adjust seasoning with salt and pepper. Serve immediately, garnished with fresh parsley and additional Parmesan if desired.