Add 2 tablespoons of butter to the pot, then add the sliced mushrooms. Cook until the mushrooms are golden brown and most of the liquid has evaporated, about 10 minutes.
Stir in the remaining 2 tablespoons of butter and sprinkle the flour over the mushrooms. Cook, stirring constantly, for 2-3 minutes to make a roux.
Slowly pour in the broth while stirring constantly to prevent lumps. Bring the soup to a simmer and cook for 10-15 minutes until slightly thickened.
Reduce the heat to low and stir in the heavy cream, thyme, nutmeg, and chopped parsley. Simmer for an additional 5 minutes.
Season with salt and pepper to taste. If the soup is too thick, you can add more broth to reach your desired consistency.
Serve hot, garnished with a sprinkle of fresh parsley and a drizzle of olive oil. Enjoy with crusty bread or croutons.