Cook the vegetables.

Heat the olive oil in a large stockpot set over medium heat. Once the oil is glistening, add the onion, carrot, and celery and cook, stirring occasionally, until the vegetables are beginning to soften, about 8 to 10 minutes. Add the garlic and cook until fragrant, about 1 more minute.

Make the roux and add stock.

Add the butter to the pot. Once the butter is bubbling, sprinkle in the flour and stir until a thick paste forms. Cook for about 1 minute. Slowly add the chicken stock, stirring constantly, and bring to a boil over high heat. As soon as the soup is boiling, reduce the heat to a simmer and cook until the vegetables are tender, and the soup has thickened, about 5 more minutes.

Finish the soup.

Add the milk, salt, pepper, dill, chicken, and Parmesan cheese, and stir to combine. Simmer until the chicken is warmed through and the Parmesan is fully incorporated, about 3 more minutes. Taste and adjust seasoning as needed.

Serve warm.

Ladle the soup into bowls and top with green onions, more dill, and extra Parmesan, if desired.