Heat olive oil in a large Dutch oven or stockpot over medium-high heat. Season diced chicken with 1/2 tsp salt and 1/4 tsp pepper. Add chicken to the pot and cook until lightly browned and cooked through, about 5-7 minutes. Remove chicken with a slotted spoon and set aside, leaving any rendered fat in the pot.

Reduce heat to medium. Add butter to the same pot. Once melted, add chopped onion, carrots, and celery. Cook, stirring occasionally, until vegetables have softened, about 8-10 minutes. Add minced garlic and cook for another minute until fragrant.

Sprinkle the all-purpose flour over the sautéed vegetables. Stir constantly for 1-2 minutes to create a roux, cooking out the raw flour taste. It should form a thick paste and become lightly golden.

Gradually whisk in the chicken broth, a cup at a time, ensuring no lumps form. Bring the soup to a gentle simmer, stirring continuously. Add the dried thyme and bay leaf. Reduce heat to low, cover, and simmer for 10-15 minutes to allow the flavors to meld, stirring occasionally to prevent sticking.

Stir in the milk (or half-and-half) and the reserved cooked chicken. Increase heat slightly and bring the soup back to a gentle simmer. Avoid a rolling boil after adding dairy to prevent curdling.

Add the wide egg noodles to the soup. Cook according to package directions, typically 7-10 minutes, or until the noodles are tender, stirring occasionally to prevent them from sticking together or to the bottom of the pot.

Remove and discard the bay leaf. Taste the soup and adjust seasoning with additional salt and freshly ground black pepper as needed. Stir in the fresh chopped parsley just before serving.

Ladle hot soup into bowls and serve immediately. Enjoy your hearty creamy chicken noodle soup!