In a large pot over medium heat, melt the butter. Add the onion and cook until tender, about 5-7 minutes.
Whisk in the flour and cook until golden, about 2 minutes. Slowly whisk in the milk.
Add in the broth, spices, and mustard. Season with salt and pepper to taste.
Reduce heat to medium low and cook uncovered for about 10 minutes.
Add in the broccoli and carrots. Simmer until tender, about 20 minutes more.
Remove the bay leaf, and puree until smooth using a stick blender.
Stir in the cheese, whisking over medium low heat, until melted. (If too thick, add a bit of water).
Serve with more cheese and parsley.
