In a large heavy-bottomed pot, melt 1 tablespoon of the butter and saute the onion until translucent. Add remaining butter and cook until melted. Stir in flour and continue cooking for 3-4 minutes. Gradually whisk in half-and-half and stock and let simmer for 20 minutes.
Reduce heat to medium-low and add broccoli and carrots. Let simmer until vegetables are tender, about 25 minutes.
Add nutmeg, salt and pepper, and cheddar cheese. Once cheese is melted, puree soup to desired consistency. (Puree all for a smooth soup, half for a chunkier soup, or none at all for a very chunky soup.) Top with a little more cheese, serve, and enjoy!