BBQ shrimp New Orleans style is prepared with the heads on and simmered in a butter sauce. This process produces a smokey red, or lightly barbecued color sauce that is very tasty. No, you do not have to eat the head, but you do need to cook it with the head on to make a proper sauce. You can simply pinch the heads off as you eat. They are divine alone or paired with crusty bread and a hearty wine.