Melt butter in a large pot over medium heat.
Add chopped onion and sauté until translucent, about 5 minutes.
Stir in minced garlic and cook for an additional minute.
Add broccoli florets and vegetable broth; bring to a boil.
Reduce heat and let simmer for 15 minutes until broccoli is tender.
Use an immersion blender to puree the soup until smooth.
Stir in heavy cream and heat through for 5 minutes.
Gradually add shredded cheddar cheese, stirring until melted.
Season with salt, black pepper, and cayenne pepper (if using).
Serve hot, garnished with extra cheese or broccoli if desired.