> 1/2" dia. ...glass, 0.004" thick
125:1 slenderness, you gonna hafta support the heck out of it.
I'm thinking metal-lathe.
Chuck a 1.5" rod of Al or Brass, face the end very flat. Mount a stick of pine in the tool holder. Wax or Balsam your glass to the rod and while hot, rotate the chuck while working the pine over the glass to get it very parallel to the rod (perpendicular to the rotation). Cool.
Now mount a diamond ring in the toolholder. Actually, a carbide-tip tool should rub glass away. It may not be ideal economics for mass production but you are just a few parts. And a carbide tool will mount a lot easier than the diamond ring.
Aim the tip just outside the 0.25" radius, turn chuck slowly, feed tool into the glass very slowly. You want to hear some action (it may be ear-splitting), but you can't afford what a metal-worker would call "pressure". Just keep rubbing until you get through the glass.
If the glass is not perfect perpendicular, it will go-through at one part while another still has thickness, and probably shatter. I hope your blanks are cheap.
Alternately: face two very short 0.5" diameter rods, center-drill the backs. Sandwich the glass between them. Mount between lathe centers with high force. Now you may be able to nibble the bulk and file-down to finish, with the part you want to keep always well-supported.
Trying to do it without a good metal-lathe.
VERY good zero-wobble drill-press. Find a 9/16" carbide-tooth hole-saw which is really 1/2" inside. Wax the glass to hard oak or flat aluminum/brass. Clamp well. Feed gently.
Same as above except steel tubing with 0.5" ID. Build a dam around the glass. Fill the pond with toothpaste or auto valve-grinding compound. Let the steel work the abrasive against the glass.
Any wobble in the drill-press will at least give an undersize result, and may just break glass.
40 years ago there was a fine air-abrasive tool which, in the ads, would cut a neat square out of an egg shell. Also, of course, various watch-size products which pay better than egg-squares. I guess it was a tiny sand-blaster, and what we now know about abrasive dusts, it may no longer be available, though it may have re-birthed as this water-jet.
Very brave/brutal approach: wax the glass between two 0.5" rods, hammer the rod through a 0.5" hole. Anything outside 0.5" will be sheared off. Maybe most of what is inside 0.5" will survive. Same as a hole-punch, except because glass has low tensile strength and elaticity you support the leading-face too.