In a large pot, heat the olive oil over medium heat. Add the chopped onion sauté until softened, about 5 minutes.
Add the broccoli florets and minced garlic, to the pot, and sauté for another 3 -4 minutes.
Pour in the vegetable broth, and bring the mixture to a boil. Reduce the heat to low, cover, and simmer for about 15 minutes, or until broccoli is tender.
While the soup is simmering, prepare the pesto. In a food processor or blender, combine the basil leaves, pine nuts, garlic, nutritional yeast, lemon juice, olive oil, salt. Blend until smooth.
Once the soup is ready, remove it from the heat and pesto. Allow it to cool slightly. Then, use an immersion blender to blend the soup until smooth. Alternatively, you can carefully transfer the soup to a blender and blend in batches until smooth.
(optional) reserve a small portion of the pesto for serving the soup
Taste the soup and adjust the seasoning with salt and pepper as needed. Ladle the soup into bowls, and drizzle each serving with a spoonful of the remaining pesto.