Melt the butter in a 2-3 quart dutch oven over medium heat. Add onion, carrot, and ¼ tsp salt. Saute until onions are translucent, about 6 to 10 minutes. Add garlic and cook for 1 to minutes more.

Adding vegetable stock, half and half, and milk. Be sure to stir to keep the milk from scorching

Add the broccoli, mustard powder, pepper, and nutmeg. Bring to a simmer, cover, stirring occasionally until tender, about 20-30 minutes. If using frozen broccoli, check to see if it's tender after 10 minutes. Add xanthan gum.

Remove from heat. Puree the soup with a stick blender or blend to the desired consistency. Remove some pieces of broccoli and carrot before pureeing if you want a chunkier soup.

Place the soup back on the heat, turned to low. Slowly whisk in the cheese a ¼ cup at a time, letting each addition melt completely before moving to the next addition.

Taste the soup add the remaining ¼ of salt if needed, as well as pepper.