Heat olive oil in a large pot over medium heat. Add onion, carrots, celery, and garlic. Sauté until softened, about 5 minutes.
Stir in thyme, black pepper, and salt.
Add shredded turkey and chicken broth to the pot. Bring to a boil, then reduce heat and simmer for 15-20 minutes.
In a separate saucepan, melt butter over medium heat. Whisk in flour and cook for 1 minute, stirring constantly.
Gradually whisk the roux into the soup, stirring until smooth.
Stir in heavy cream and milk. Bring to a simmer, stirring occasionally, until the soup thickens.
Ladle soup into bowls and garnish with fresh parsley.
