In a large Dutch oven, melt butter over medium heat.
Add the carrots, onion, celery, and garlic. Sauté the vegetables until the onions are translucent, about 5-7 minutes.
Stir in the flour, bay leaf, thyme, salt, and pepper. Cook for 1 minute, stirring constantly.
Gradually add the broth and milk, stirring constantly. Gradually bring the liquid to a low boil, stirring regularly. Once the liquid boils, reduce the heat so that it’s just simmering. Discard the bay leaf and thyme sprigs (if using fresh thyme).
Add the noodles to the pot, and simmer, stirring regularly, until the noodles are tender, about 5 minutes.
Reduce the heat to low. Add the cooked chicken, half-and-half or heavy cream, and parsley. Stir over low heat, just until warmed through. Discard bay leaf. Taste and season with additional salt and pepper, if necessary.