Melt 4 tablespoons of butter in a large stockpot over medium-high heat.
Add onion, carrots, celery, garlic, salt, pepper, and nutmeg.
Cook, stirring often, until the carrots are tender, about 10-15 minutes.
Stir in additional butter and allow to melt.
Sprinkle in flour and whisk for 60 seconds.
Slowly whisk in broth, then stir in the half and half.
Bring to a simmer, stirring regularly, then stir in broccoli.
Cover and simmer for 20-30 minutes (you’ll likely need to lower your heat to keep at a simmer) or until broccoli is tender, stirring occasionally to prevent burning.
Turn off the heat, then stir in the cheese little by little until melted.
Taste and re-season, if necessary.
Serve and enjoy!
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