In a large soup pot, heat the lard on medium-high heat. Add garlic, onions, and celery and saute until soft.
Add broth, broccoli, salt, thyme, and nutmeg. Bring to a simmer and cook until broccoli is quite soft.
Remove from heat and add coconut milk. Adjust seasonings if necessary.
Use hand blender to lightly break up the bigger chunks of broccoli. I prefer my cream of broccoli soup a little on the chunkier side, but if you like it smoother, puree a little longer. This step is optional