Add the chopped onion and cook until softened, about 5 minutes.
Add minced garlic and cook for another minute, or until fragrant.
Pour in vegetable broth and add chopped broccoli florets. Season with salt and pepper.
Bring to a boil, then reduce heat and simmer for 10–15 minutes, or until broccoli is tender.
While the broccoli is simmering, heat the milk in a separate saucepan over low heat. Do not boil.
When the broccoli is tender, use an immersion blender or carefully transfer the soup in batches to a regular blender to puree until smooth.
Return the pureed soup to the pot and stir in the warm milk.
Gradually add the shredded cheddar cheese, stirring until it is melted and the soup is creamy.
Taste and adjust seasonings, if necessary.
Serve warm with crusty bread or crackers.
Notes
If you don’t have an immersion blender, you can use a regular blender. Be sure to let the soup cool slightly before blending, and vent the blender lid to release steam.
If you prefer a chunkier soup, you can mash some of the broccoli with a fork instead of pureeing all of it.
For a richer soup, you can use heavy cream instead of milk.