Why not barbecue itself? There are very few times when one will associate barbecue without fire (or grill) or food.
Note that barbecue doesn't necessarily specify the method (grilling, broiling, etc.) of cooking, just that it's a [gathering around a] heat source for cooking out of doors.
The answer is to the title. But the body asserts a different question:
The barbecue tasted incredible, with that unique aroma that only [...] gives to the meat.
Except, generally, wouldn't you want to eat an item prepared on the barbecue?
The chicken tasted incredible, with that unique aroma that only barbecue gives to the meat.
This would present a better picture overall, because it not only describes what is being cooked, but how it was prepared.