In a large pot over medium heat, melt the butter and olive oil.
Add the garlic and cook for 1-2 minutes, or until fragrant.
Add the chopped broccoli, chicken or vegetable broth, heavy cream, cheddar cheese, Parmigiano Reggiano, salt and pepper to the pot.
Bring to a boil, then reduce the heat and simmer for 20-25 minutes, or until the broccoli is tender.
Use an immersion blender or transfer the soup to a blender and blend until smooth.
Taste and adjust seasoning as needed.
Serve the soup warm, garnish with additional cheese and/or chopped herbs, if desired.