Broken screw

I think the answer is probably "you're not", at least not without breaking something.

My approach would be (if it really *needs* to come out) to try and remove most/all of the head using a drill or dremel or similar, then remove the coat hanger and finally remove the stub of the screw using a mole wrench or similar.

One method is to use a "Dremel-type" tool with a grinding bit to remove the remaining part of the head. The coat hook can then be taken off and the screw removed using a mole-grip on the remains of the screw shaft.

Another is to take a steel bar of the same diameter as the screw head (or just use another screw) and grind a part off the end so that it will fit into the countersink, overlapping the remaining part of the screw. Put the bar in a brace, so that a lot of pressure can be applied, and turn the screw out enough so that it can be mole gripped.

Another is to use a "Dremel-type" tool, with a very small diameter drill bit, to drill a very small hole partly in the head and partly in the hook. Then hammer a wire nail into the hole. The hook will now be locked to the screw, so turning it will wind-out the screw...

Another is to use a screw extractor - it helps if you can centre-pop the screw head to give the extractor somewhere to locate in..

Thanks for the advice