In a large pot or Dutch oven, heat the olive oil or melt the butter over medium heat. Add the chopped carrots, celery, and onion. Sauté for 5-7 minutes, or until the vegetables are softened and the onion is translucent. Add the minced garlic and cook for another minute until fragrant.

Sprinkle the all-purpose flour over the sautéed vegetables. Stir constantly for 1-2 minutes, allowing the flour to cook and form a roux.

Gradually whisk in the chicken broth, ensuring no lumps form. Bring the mixture to a gentle simmer, stirring occasionally. Add the dried thyme and rosemary, along with salt and freshly ground black pepper to taste. Continue to simmer for 10-15 minutes, allowing the flavors to meld and the soup to thicken slightly.

Add the egg noodles to the simmering soup. Cook according to package directions, usually 7-10 minutes, or until the noodles are tender. Be careful not to overcook the noodles, as they can become mushy.

Reduce the heat to low. Gradually stir in the milk or half-and-half. Add the shredded or diced cooked chicken. Heat through gently, stirring occasionally, for about 5 minutes. Do not bring to a rolling boil after adding the milk, as it can curdle.

Taste the soup and adjust seasonings as needed. You might want to add more salt, pepper, or herbs to suit your preference.

Ladle the hot soup into bowls. Garnish with fresh chopped parsley before serving.