Place broccoli florets and carrots in a dutch oven or soup pot and cover with water. Bring to a boil over medium-high heat and boil for 3 minutes. Drain vegetables in a colander and set aside.
In the same empty soup pot, melt butter over medium heat. Add onions and cook until tender, about 3 minutes (stirring frequently).
Slowly add flour, about a tablespoon at a time, while constantly stirring. Once all of the flour is added and there is a paste formed, slowly add chicken broth, while continuing to stir.
Add milk slowly, stirring constantly. Bring to a gentle boil over medium heat, stirring constantly.
Slowly add cheese, continuing to stir until cheese is melted.
Add broccoli and carrots, garlic, nutmeg, salt, pepper and bay leaf. Stir well to combine. Reduce heat to low. Cover and cook over low heat for 20 minutes, stirring occasionally. Remove and discard bay leaf before serving.