In a large pot or Dutch oven, melt butter over medium heat.
Add onions and cook for 5 minutes.
Add garlic and cook for another minute.
Add flour and stir to coat, then add milk and stir to thicken.
Add broth, carrots, broccoli, Dijon mustard and black pepper. Heat to a boil, then turn down heat to a simmer for 15-20 minutes.
Stir in the cheese. Add more broth to thin out as needed.
