Begin by melting the butter in a large pot over medium heat. Add the chopped onion and sauté until translucent, about 5 minutes.
Stir in the minced garlic and cook for an additional 1-2 minutes, until fragrant.
Add the broccoli florets to the pot and pour in the vegetable broth. Bring to a gentle boil, then reduce the heat and let it simmer for 15-20 minutes, or until the broccoli is tender.
Once the broccoli is cooked, use an immersion blender to purée the soup until smooth. You can also carefully transfer the soup to a blender in batches if you don’t have an immersion blender.
Return the soup to the pot and stir in the heavy cream, dijon mustard, smoked paprika (if using), and grated cheddar cheese. Mix well until the cheese is melted and the soup is creamy.
Season with salt and pepper to taste, adjusting as necessary. Allow the soup to heat through on low heat for another 5 minutes.
Serve hot, garnished with fresh chopped parsley on top and extra cheddar cheese if desired.