Melt the butter in a large pot over medium heat. Add the onion, carrots, celery, mushrooms, and garlic. Cook, stirring occasionally, for 5 minutes or until vegetables have softened and released some liquid. Stir in flour, salt, pepper, and thyme until flour has absorbed all the liquid. Stir in the broth and milk. Cook and simmer for 7–9 minutes or until thickened into a thick soup-like consistency. Stir in the green peas and parsley and then remove from heat. Taste and add more salt, pepper, thyme, or parsley if desired. Cool for 5 minutes.