Melt the butter in a Dutch oven over medium-high heat. Add the onion, carrots, and celery. Season with the salt and pepper and cook 5 minutes or until the onions are tender. Add the flour and stir to coat the vegetables evenly.
Gradually add the chicken broth and the milk, stirring to combine. Add the bay leaf and thyme sprigs. Bring the soup to a simmer.
Remove the dumplings from the freezer while bringing the soup to a simmer (you'll only need half the package of dumplings). Carefully tear the dumplings along the perforations. Add 8 to 10 dumplings to the simmering soup.
Return to a boil over medium heat. Add 8 to 10 more dumplings. Continue the process of letting the soup come to a boil and adding the dumplings until all the dumplings are added. Cook for 10 more minutes or until tender. Remove the bay leaf and thyme sprigs from the soup.
Stir the chicken into the simmering soup and cook until heated through, about 5 minutes. Season with additional salt and pepper to taste if necessary.
To serve, divide the soup and dumplings evenly between serving bowls. Garnish with parsley, if desired.