In a large pot or dutch oven, melt the butter over medium-high heat. Add the onion and carrot. Cook until slightly softened, about 5 minutes.
Add garlic, Italian seasoning, salt, mustard, and pepper. Cook to toast spices for 1 minute.
Stir in flour and cook for 1 minute.
Add vegetable broth, milk, and cream. Cook and stir until it begins to thicken.
Add broccoli to soup, reduce to medium-low heat, and simmer until tender (approximately 10 minutes).
Add shredded cheese and stir until the soup is smooth. Serve.
