In a large pot, melt butter over medium heat. Sauté the onion, carrots, celery, and garlic for 5 minutes or until softened.
Sprinkle flour over the vegetables and cook for 1-2 minutes, stirring constantly.
Gradually whisk in chicken broth, ensuring there are no lumps, and bring to a simmer.
Add the thyme, parsley, salt, and pepper, then stir in the shredded chicken and pasta. Simmer for 10-12 minutes, until pasta is cooked.
Lower the heat and stir in milk or cream. Heat through but do not boil.
Serve hot, garnished with fresh chives.