In a large stockpot melt 4 tablespoons of unsalted butter over medium heat. Add the onions and carrots, sauté until tender about 5-10 minutes. Add the garlic and sauté for an additional 2-4 minutes. Add the flour and stir until golden. Add the chicken stock and Half & Half, stir until smooth.
Add the chopped broccoli and seasonings; salt, pepper, dry mustard, and paprika, stir to combine. Bring to a low boil, then reduce heat to low and simmer for 15 to 25 minutes until broccoli is tender. Turn the heat off and add the cheddar cheese, stirring until melted throughout. Adjust salt and pepper to taste.