In a large pot or Dutch oven, melt the butter over medium heat. Sauté the chopped onion until it becomes fragrant, about 2 minutes. Add the garlic and cook for another minute. Sprinkle in the flour and whisk constantly, cooking for a few minutes until it turns golden brown.
Reduce the heat to medium-low and gradually add the chicken stock and half-and-half, stirring continuously to incorporate the flour into the liquid. Season with salt, pepper, stock powder, mustard powder, and garlic powder. Stir well and let the mixture cook and thicken for approximately 5 minutes, stirring occasionally.
Add the broccoli florets and grated carrots to the pot. Allow the soup to gently simmer for another 20 minutes, until the broccoli is fully tender. Stir in the cheddar cheese until just melted and well combined. Taste and adjust the seasoning with additional salt and pepper if necessary.
Serve hot with bread rolls for a hearty meal.
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