Heat a large pot over medium heat. Add 4 Tablespoons of butter and let it melt.
Add onions and carrots and cook for about 7-8 minutes, stirring often.
Add garlic and saute for 1 minute.
Add remaining 4 Tablespoons of butter. Stir in flour.
Add chicken broth and use a whisk to mix together.
Add dried mustard powder, cayenne, salt, and pepper.
Add broccoli and half-n-half. Let simmer for about 20 minutes or until broccoli is softened.
Add cheese ½ cup at a time, stirring well after each addition. This helps to keep the creamy consistency and not end up with a grainy consistency.
Taste for seasonings. Add more salt and pepper, if needed.
Serve with homemade garlic butter croutons or buttery crusty bread for dipping.
