Melt butter in a large pot over medium heat. Saute diced onion for 5-6 minutes until translucent.
Add cayenne pepper, minced garlic, and grated carrots. Cook for 7-8 minutes until softened.
Sprinkle in flour and cook for 2 minutes, stirring constantly.
Gradually add chicken broth while stirring. Add chopped broccoli and simmer 8-10 minutes until tender.
Pour in the half and half. Bring to a gentle simmer.
Turn off heat and stir in shredded cheddar cheese until fully melted and smooth.
Season to taste and serve hot with crusty bread.