In a large pot, melt the butter over medium heat. Add the diced onion and sauté for about 5 minutes until softened, stirring occasionally.
Stir in the minced garlic and cook for an additional minute until fragrant.
Add the broccoli florets and vegetable broth to the pot. Bring the mixture to a boil, then reduce the heat to a simmer. Cover and cook for about 10-12 minutes, until the broccoli is tender.
Remove the pot from heat. Use a blender or immersion blender to puree the soup until smooth. If you prefer a chunkier texture, blend only half of the soup or leave it unblended.
Return the pot to low heat and stir in the heavy cream. Slowly add the shredded cheddar cheese, stirring until melted and well combined.
Season the soup with salt, pepper, and nutmeg if using. Taste and adjust seasoning as desired.
Serve hot, garnished with additional cheddar cheese if desired.