The name probably derives from the handyman's responsibility for preparing and heating ("cooking") the bran mash fed to the bulls (oxen) at a lumber camp.
1913  Bull-cook, a handyman in a lumber camp, who carries water, splits wood, and does other chores, especially assisting the cookees.
1947  The bullcook will be reminded to keep the sinks as clean as possible.
1963  But even such lowly creatures as laborers and bull cooks and chain men followed the progress of the steel.