Melt butter in a large soup pot over medium-high heat; cook mushrooms in butter with 1 pinch salt until the mushrooms give off their juices; reduce heat to low. Continue to cook, stirring often, until juices evaporate and the mushrooms are golden brown, about 15 minutes. Mix onion and garlic into mushrooms and cook until onion is soft and translucent, about 5 more minutes.

Stir flour into mushroom mixture and cook, stirring often, for 2 minutes to remove raw flour taste. Pour chicken stock and water into mushroom mixture. Add thyme in. Bring to a simmer and cook for 1 hour.

Purée soup until smooth and thick. Add half and half, salt & pepper, and Parmesan cheese and stir altogether.