Melt butter in a large non stick pot. Add the onions. Cook for 2-3 minutes until lightly golden.
Stir in the garlic and cook for a further minute. Then stir in the flour and cook for 2 minutes.
Pour in the stock and milk. Season with salt and pepper. You will need less salt then usual based on the stock - about 1/2 - 3/4 tsp. Bring to the boil, then simmer for 5 minutes.
Add the broccoli and carrots. Simmer for 15 - 20 minutes, blend using a stick blender, then stir in the cheese. Check seasoning and adjust if need be.