My way:
Split out, or bandsaw, a straight grained blank roughly appropriate in thickness and about 6" longer than finished size. Set it aside to dry -over time, or use the oven, carefully. Have your hawk available; check the eye for possible taper and profile , to determine whether it will be slip mounted or wedged. Visualize the finished handle. Orient the growth rings [radials] to run in line with the blade and pole, and mark off any contours needed or desired. Mount the blank in a vise if you don't have access to a shaving horse and shape it with drawknife and/or spokeshave, reversing as needed. If the head is to be wedged, you might find it helpful to final fit using a rasp or coarse file. Allow the haft to pertrude thru the head a bit when fitted, leaving about a half inch if you're slip- fitting the head. I prefer using a full width softwood wedge inserted and glued in a saw kerf cut parallel with the growth rings. Cut off the excess lenght at the butt end -and at the head, if wedged, smooth the full lenght with a scraper or knife edge and rub in a generous coat of boiled linseed oil.
Might as well make a spare if its a throwing hawk.
~Longshot