In a large pot, sauté the onion in 1 tablespoon melted butter until onions are soft. Add in remaining butter and flour. Stir with the onions over medium heat for about 3-4 minutes. (yes, it's a roux starter)
Slowly whisk in the half and half and chicken stock. Let it simmer for about 20 minutes.
Add the broccoli, carrots, and onions. Let them simmer on medium low 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. For a smoother soup you can puree it in a blender (But we love the chunkier texture!)