In a large saucepan over medium heat melt butter and saute onion and green bell pepper until softened. Add 1/4 cup butter to the mixture, and stir until melted. Add flour and garlic powder; stir and cook for 1 minute. Slowly add the half-and-half, stirring constantly. Stir in the chicken stock; reduce heat to medium-low and simmer for 20 minutes. Add the broccoli, carrot and pimientos; simmer for another 10 minutes or until vegetables are softened. Add the cheese, and stir until melted and smooth. Season to taste with salt and pepper.