Add flour and whisk for 1-2 minutes or until the flour begins to turn golden in color. Pour in the chicken stock, broccoli florets, carrots, and seasoning. Bring to a boil then reduce heat to medium-low and simmer for 15 minutes or until the broccoli and carrots are cooked through.
Stir in half & half and cheddar cheese and simmer for another minute. Taste, and adjust seasoning if needed.
Serve with warm dinner rolls or in a bread bowl if desired.