In a large pot, melt the butter over medium heat. Add the chopped onion and sauté until translucent, about 5 minutes. Stir in the minced garlic and cook for an additional minute until fragrant.

Add the broccoli florets to the pot and pour in the vegetable broth. Bring to a boil, then reduce the heat and let it simmer for about 10-15 minutes, or until the broccoli is tender.

Using an immersion blender, carefully blend the soup until smooth. If you don’t have an immersion blender, you can transfer the soup in batches to a regular blender.

Return the blended soup to the pot over low heat. Stir in the heavy cream and shredded cheddar cheese, mixing until the cheese is fully melted and the soup is creamy.

Season with salt, pepper, and Dijon mustard if using. Heat through for another 5 minutes, stirring occasionally.

Serve hot, garnished with chopped chives.