Prepare the vegetables by trimming the woody ends off the broccoli stems (about 1 inch). Chop the rest of the broccoli stems into thin slices (about 1/8th inch) and cut the broccoli florets into bite sized pieces. Dice the onion and slice the carrot into 1/4" pieces.
Melt butter in a medium-sized heavy pot.
Add broccoli, onion, carrot, salt and pepper to pot and saute until onion is translucent, about 6 minutes.
Add xanthan gum to pot with vegetables and cook for 1 minute.
Add chicken stock, salt and pepper to pot and bring to a boil. Simmer mixture, uncovered, until broccoli is tender, about 15 minutes. A fork should be able to slide easily in and out of the broccoli.
Add half and half to pot.
Blend soup in the pot with an immersion blender, or blend in batches in a blender. Be careful as the soup will be hot! I prefer some texture in my soup, but blend to desired consistency.
Add additional salt, pepper, and red pepper flakes (if using) to taste.