Melt the butter in a heavy pot over medium heat and cook the onion for 5 minutes until soft but not browned.
Stir in the flour and cook for 1 minute until it smells lightly toasted.
Whisk in the broth and milk a little at a time, then simmer for 5 minutes until the base coats the back of a spoon.
Add the broccoli and carrot and cook for 8 to 10 minutes until the broccoli is tender with a little bite.
Take the pot off the heat, stir in the cheddar by handfuls until smooth, then season with the salt and nutmeg.