Not sure what kind of wood this is, but it’s a good match for the old handle.
Center drilled the square blank with 1/4″ bit, then mounted it on a pen mandrel. Need to use non-standard bushings, as the Slimline bushings that came with it are just small enough to press-fit into the blank.
After shaping and finishing with friction polish, I opened the hole to the correct diameter using a Rockler pen vise. I started with a size P drill, then drilled the final size.
Unfortunately there was a little damage when it spun in the vise while I was still figuring out the procedure. Next time I need to remember to loosely clamp it and then hold it with my hand to keep it from spinning.