Melt the butter in a Dutch oven over medium heat and cook the onion and carrot for 6 minutes until soft.
Sprinkle in the flour and stir for 2 minutes until the paste smells lightly nutty.
Whisk in the broth gradually, then add the milk and simmer for 8 minutes until lightly thickened.
Add the broccoli, salt, black pepper and nutmeg and cook for 10 to 12 minutes until the florets are tender but still green.
Remove the pot from direct heat and add the cheddar a handful at a time, stirring until smooth.
Return to low heat only if needed and warm to 165°F without boiling.