In a large pot, melt the butter over medium heat. Add the diced 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 heat and let it simmer for about 15 minutes, until the broccoli is tender.
Once the broccoli is cooked, use an immersion blender to puree the soup until smooth. If you prefer a chunkier texture, blend only half or use a regular blender in batches.
Return the pot to low heat, then stir in the heavy cream, grated cheddar cheese, mustard powder, salt, pepper, and cayenne pepper if using. Stir until the cheese is fully melted and the soup is creamy.
Taste and adjust seasoning if necessary. If the soup is too thick, you can add more vegetable broth to reach your desired consistency.
Serve hot, garnished with croutons if desired.
Notes
Add more vegetable broth for a thinner consistency.