Finely chop the onion, carrots, and celery. Mince the garlic. Chop broccoli into 1/2-inch florets, and finely chop any broccoli stems. Set garlic and broccoli aside.

Melt the butter in a Dutch oven over medium heat. Add the onion, carrots, celery, and salt. Saute until vegetables are softened, 6-7 minutes.

Add the minced garlic and mustard and stir until fragrant, about 1 minute. Sprinkle the flour over vegetables and stir to evenly coat.

Stream the broth and milk into the pot, stirring to dissolve the flour. Season soup with the salt and pepper, then add broccoli.

Bring the soup to a low boil, stirring occasionally. Reduce heat to low, cover, and simmer, stirring once or twice, until broccoli is just tender, about 10 minutes.

Turn off the heat, but leave pot on the stove. Add half of the cheese in two additions, stirring after each, until completely melted.

Mince the parsley. Ladle soup into bowls and sprinkle with parsley. Add salt and freshly ground pepper to soup if desired.