Heat olive oil in a large pot over medium heat. Add the onion and cook for 3-4 minutes, until softened.
Add the garlic and cook for 1 more minute. Add the broccoli florets, potatoes, and vegetable broth to the saucepan. Bring to a boil. Then reduce to a simmer and cook for 20-30 minutes, until broccoli and potatoes are fork tender.
Use an immersion blender to puree the soup to a creamy consistency. If you do not have an immersion blender, transfer the soup to a regular blender for this step, and then return it to the pot.
Stir in ½ cup of the parmesan cheese, and stir until well combined.
Serve sprinkled with fresh parsley and the remaining parmesan cheese.
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