Melt the first tablespoon of butter in a saucepan on medium heat.
Add onion to the pan and sauté until the onion becomes translucent.
In another saucepan, add remaining butter and melt on medium heat. Add flour and whisk for five minutes. It should turn golden brown by this point.
Gradually pour chicken stock into the flour and butter mixture and simmer for 20 minutes with lid on the pan.
Add carrots, broccoli, and onions, milk, and half-and-half to the pan. Stir and reduce heat to low.
Put the lid back on and let heat for 20 minutes (do not bring to boil).
Add seasonings - nutmeg, salt, and pepper and stir.
Gradually add cheese a little at a time. Stir until melted.
Serve with bread and refrigerate leftovers.