Melt the butter in a large pot over medium heat.
Add the diced onion and cook until soft, about 5 minutes.
Stir in the minced garlic and cook for an additional minute until fragrant.
Add the broccoli florets and stir to combine with the onion and garlic mixture.
Pour in the vegetable broth and bring to a simmer.
Simmer the soup for 15-20 minutes until the broccoli is tender.
Using an immersion blender, carefully blend the soup until smooth.
Stir in the heavy cream and shredded cheddar cheese.
Season with salt and pepper to taste, then serve warm.