Melt the butter in a large pot over medium heat.
Add the chopped onion and cook until it's translucent, about 3-4 minutes.
Sprinkle the flour over the cooked onion.
Stir everything thoroughly for 1-2 minutes, letting the flour cook a little bit.
Pour in the chicken broth and half-and-half.
Bring the mixture to a gentle simmer while stirring frequently.
Add the broccoli to the pot.
Cook until the broccoli is tender, which usually takes around 15-20 minutes.
Stir in the shredded cheddar and nutmeg.
Season with salt and pepper.
Serve hot and enjoy!