In a large Dutch oven or pot, melt the butter over medium heat. Add the chopped onion and sauté until translucent, about 5-7 minutes.

Add the minced garlic and sliced mushrooms to the pot. Cook, stirring occasionally, until the mushrooms have released their liquid and started to brown nicely, about 10-12 minutes. Stir in the fresh thyme.

Sprinkle the flour over the mushrooms and stir constantly for 1 minute to cook out the raw flavor of the flour, creating a roux base.

Slowly whisk in the vegetable or chicken broth, ensuring no lumps form. Bring the mixture to a gentle simmer, then reduce heat to low, cover, and cook for 10 minutes to allow flavors to meld.

Carefully transfer the soup mixture (reserving about 1/4 cup of whole mushrooms if desired for texture) to a blender or use an immersion blender directly in the pot. Blend until completely smooth and creamy. Return the pureed soup to the pot if using a standard blender.

Stir in the heavy cream, Worcestershire sauce, and the grated Parmesan cheese until the cheese is fully melted and incorporated. Heat through gently, but do not boil once the cream is added.

Taste the soup and adjust seasoning with salt and freshly ground black pepper as needed. Serve hot, garnished with extra grated Parmesan and fresh parsley.