Doc Green
Here's one way to do it that doesn't require a collet chuck, steady rest, or anything else out of the ordinary.

1. With the handle mounted between centers, turn a short tenon on the back end of the handle.

2. Remove the handle from between centers. Use a countersink mounted in a hand drill to enlarge the point where the tail center was. The idea is to make certain the drill bit used to bore the hole finds the center. (You may skip this step if you use a spade bit or Forstner bit.)

3. Install the drill bit in a Jacobs chuck mounted in the tailstock.

4. Mount the handle in a scroll chuck using the tenon you created in step 1.

5. Bring up the tailstock, drill bit, etc., and seat the end of the bit in the recess you made with the countersink in step 2. Put just enough pressure on the piece to hold it in place temporarily.

6. Make sure the lathe is set for a low RPM, perhaps 400 or so. Start the lathe and drill the hole by advancing the quill of the tailstock. When you reach the desired depth, turn the lathe off, then pull the bit out. Remove the handle from the lathe.

7. Reverse the handle to remove the tenon you cut in step 1. Make a jam chuck with a tenon the same size as the hole you drilled for the tool. Slip the handle over the tenon and bring up the tailstock to hold everything in place. Remove the tenon from the end of the handle by turning it down to a nubbin, and so forth. And that's it.

Caution: Some brad point drill bits will tend to self feed and screw themselves into the wood without making a hole when turned slowly with considerable pressure applied. My suggestion is to use a conventional drill bit or even a good-quality spade bit. If you use a Forstner bit, it will need to be sharp.

And, as always, there are other ways to do it. Good Luck. ~Doc