Hi aaa,
You have to check per clinic as they will have their own protocols obviously. In most cases it is normal practice for the doctor to punch all the grafts in the donor, they and sometimes their FUE technician will then remove the grafts from the donor. The doctor would then be making the placement sites in the recipient, hair line design etc and then the placement techs do the placing, the doc may place some of the case but not on every case by far, that is tech work. Some doctors do the entire procedure by themselves, this is obviously going to be slower with fewer graft numbers achieved per session, also you have to put into the equation fatigue of one person doing the entire procedure.
It is important you talk to the clinic first and get their protocol so you don't have any surprises on the day; also look at their track record for results, grown out cases and previous patients to get an idea.