Heat olive oil in a large stockpot or Dutch oven over medium heat. Season chicken thighs with salt and pepper, to taste. Add chicken to the stockpot and cook until golden, about 2-3 minutes; set aside.
Melt ghee in the large pot over medium heat. Add mushrooms, onion, garlic, carrots and celery. Cook, stirring occasionally, until tender and vegetables become translucent, about 3-4 minutes. Stir in thyme, about 1 minute.
Whisk in flour until lightly browned, about 1 minute. Whisk in chicken stock, bay leaf and chicken thighs, and cook, whisking constantly, until slightly thickened, about 4-5 minutes.
Stir in milk and cream until heated through, about 1-2 minutes; season with salt and pepper, to taste. If the soup is too thick, add more half and half as needed until desired consistency is reached.
Serve immediately, garnished with parsley.