Heat the olive oil and butter in a large Dutch oven or soup pot over medium heat. Add the onion, carrots, celery, and mushrooms, then cook for 7–9 minutes, stirring occasionally, until the vegetables soften and the mushrooms release most of their moisture.

Stir in the minced garlic and cook for about 30 seconds, just until fragrant. Sprinkle in the flour and stir constantly for 1–2 minutes so the flour cooks and loses its raw taste.

Slowly pour in the dry Marsala wine while scraping the bottom of the pot to loosen browned bits. Let it simmer for 2–3 minutes so the wine reduces slightly and blends with the vegetable mixture.

Add the chicken broth, chicken, kosher salt, pepper, thyme, and bay leaf. Bring the soup to a gentle boil, reduce the heat, and simmer for approximately 15–20 minutes, or until the chicken reaches 165°F internally.

Transfer the cooked chicken to a plate and shred or dice it into bite-size pieces. Return the chicken to the pot and stir to distribute it through the soup.

Stir in the heavy cream and simmer gently for 3–5 minutes. Do not let the soup boil vigorously after adding the cream, because gentle heat keeps the texture smooth.

Remove the bay leaf, taste the soup, and adjust the salt and pepper as needed. Finish with chopped Italian parsley and serve hot.