Sauté: Melt butter in a large pot over medium heat. Add onion and carrots; sauté for 5 minutes until softened.
Roux: Whisk in the flour, garlic powder, mustard powder, and paprika. Cook for 1 minute, stirring constantly.
Liquids: Gradually whisk in the broth and milk until smooth. Bring to a simmer.
Simmer: Add the broccoli florets. Reduce heat to low, cover, and simmer for 10-12 minutes, or until broccoli and carrots are tender.
Finish: Remove from heat. Stir in the grated cheese a handful at a time until melted and smooth. Season with salt and pepper.
Serve: Serve hot with extra cheese on top if desired.