Step 1

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.

Step 2

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.

Step 3

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

Step 4

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

Step 5

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.

Step 6

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

Step 7

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