Melt 10 gms butter in a wide pan. Saute the onion till it becomes soft. Add carrot and broccoli and cook for 2-3 mins. Keep aside.
In a large pot / sauce pan melt the butter (50 gms). Whisk together the melted butter and flour over low -medium heat for about 3-4 minutes.
Slowly whisk in the half and half / milk and chicken stock. Let it simmer for about 20 minutes.
Add the broccoli, carrots, and onions. Let them simmer on low-medium, for about 25 minutes until the broccoli and carrots are tender.
Add nutmeg, salt and pepper and sharp cheddar cheese. Let the cheese melt and then serve.