Melt the butter in a large soup pot. Add the chopped onion. When onion is soft and golden brown, add flour and stir, being careful to blend fully without scorching. Slowly add the cream, and then the chicken broth, bay leaves, nutmeg and salt. Simmer 20 minutes.
Add broccoli and carrot and simmer an additional 20 minutes until soft.
Let cool slightly, remove bay leaves, and transfer soup to a blender. Blend until carrots and broccoli are finely chopped or pureed.
Return soup to pot. Add cheese, and stir until melted.