Add the butter to a large heavy-bottomed stock pot over medium heat. Once the butter has melted, add the onions, celery and garlic with a pinch of salt. Cook, stirring, until the vegetables are tender, about 5 minutes.
Stir in the flour and continue cooking for an additional 2 minutes.
Stir in the milk, then stir in the stock, broccoli and ground mustard. Simmer the soup until the broccoli is tender, about 10 minutes.
Stir in half of the cheese until it has melted, then stir in the remaining cheese and continue stirring until all of the cheese has melted. Taste and season the soup with salt and pepper, then serve garnished with croutons and additional shredded cheese. (See Kelly's Note.)