I have used a piece of thin metal tubing (thin wall electrical conduit) to cut the recess. Cork cuts quite easily with just hand pressure. I found the tubing made a cleaner cut than a forstner bit.

I chucked the tubing in my lathe and used a metal file to sharpen one end to a bevel. I added some 'saw teeth' to the bevel using a triangular file, but I'm not sure this was needed. Then I drilled out a blank cork ring by hand using slight downward pressure over a cutting board. The ring was then glued onto the end of a pre-formed cork handle and shaped to size.

If the your handle is already glued up (so you can't add a new cork ring) you could use this system to cut the outer ring profile then use a dremel burr or an undersized spade bit or forstner bit to hog out the interior.