Remove florets from broccoli stems and set aside for other use. Trim stems and slice into thin coins.
Heat olive oil in a large pot and saute broccoli stems, onions, carrots and celery, about 7-10 minutes. Add garlic and saute until fragrant.
Add salt, pepper, nutritional yeast, potatoes and vegetable broth. Bring to a boil and simmer for 20 minutes.
Turn off heat and stir in 1/2 cup milk. Use an immersion blender or blender to puree the soup.
Once blended, add another 1/2 cup of milk to thin, if needed.
Add salt and pepper, to taste.
Garnish with roasted broccoli florets, fresh parsley, additional nutritional yeast for “cheesiness,” and serve with crackers.