If bourdieu had chosen to do his work in a highly specialized school—a culinary (cooking) school, for instance—what aspect of his theoretical outlook would he have been ascribing as an integral part of that work?

The answer is habitus
habitus refers to a tendency to organize ourselves so we could give a faster and proper response to the social environment around us.
In this case, by surrounding with environment that focus on enchancing cooking skill, he will more likely to develop a hhabbit that would help him to survive within that exact environment in the future.