Heat the 4 tbsps. of butter in a pan. Sweat the onion, carrot and celery with a little salt and pepper until softened, about 10 minutes. Add the garlic and mushrooms. Cook, stirring, for another minute. Take the pan off the stove and keep warm.
In a large thick-bottomed pot, make a blonde roux. Add the vegetables to the roux including any liquid from the first pan.
Stir together the milk and broth. Pour into the pan little by little, mixing as you pour. Note that the soup thickens as it cools so what might look like a thin soup at this point may not be too thin at all after you remove the pot from the stove.
Taste the soup and add more salt and pepper, as needed.
When the soup reaches a gentle boil, set the heat to the lowest setting and simmer the soup for 10 to 15 minutes. Stir occasionally, scraping the bottom of the pan. It's a good idea to taste the soup every time you stir and adjust the seasonings, if needed.
Ladle hot soup into bowls, pour in a little cream and sprinkle in a bit of chopped parsley. Serve hot.
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