If using fresh broccoli, remove the tough lower stalk but chop the other stalky parts.
Line a large pot with vegetable broth and sauté onion and garlic until onion is translucent
Add ½ cup broth and cooked rice or potato and let simmer until rice is waterlogged and extra liquid has all or mostly cooked off.
If using fresh broccoli, add another ½ cup broth and broccoli, and cook until broccoli is very tender. If using frozen broccoli, add broccoli at the end as the rice is finishing up (omit extra ½ cup broth - just a splash will do).
Transfer soup to a blender and puree with non-dairy milk, plus additional broth as necessary to achieve desired consistency. You can also add fresh or dried basil or type, plus salt and pepper to taste.
Return blended soup to your saucepan and let simmer, adding nutritional yeast to taste.
Squeeze fresh lemon juice over soup before serving. if desired, and garnish with red pepper flakes or fresh black pepper.
Notes
For a richer soup, blend some raw cashews with water to make a cashew cream and replace the nondairy milk with the cream.