Melt butter and olive oil in large stockpot or Dutch oven over medium heat. Add diced onion and carrot and cook for 5 minutes, or until onion is translucent. Add garlic and cook for another 30 seconds. Stir in flour and cook for another 1-2 minutes.
Gradually whisk in chicken broth, milk and broccoli. Bring to boil and then reduce to simmer for 15-20 minutes, or until broccoli is cooked.
Add salt, mustard powder, paprika and black pepper. Taste and adjust for seasoning.
Add cheese and whisk until melted. Transfer soup to bowls, garnish with additional cheese, and serve immediately.
Notes
Sautéing the onions, carrots and garlic before adding broth and roasted broccoli is just another additional flavor profile.
Use quality chicken broth or vegetable broth if you want it vegetarian. Broth is a huge component in the soup.
Finely chop the broccoli so you have a ton of small bites throughout and it cooks through.
Don’t use pre-shredded cheese. It doesn't taste as good and also has additional ingredients that you don't need.