Nelson, there are a few "tricks", but they run from extreme to mildly amusing.
If the workpiece is suitable, slam it down against an anvil like using an impact bullet puller. Dents must be allowed here.
Take a OX torch and red heat the workpiece which anneals the HSS and drill it out.
Use a carbide end mill and cut it out slowly.
Special grind a hollow end mill and cut around it, then chip it out with a prick punch. Think reverse countersink shape.
If the length/depth (whatever) allows, cut off enough stock to get beyond the broken bit.
If the workpiece happens to be a stainless alloy, get some Tap-Out and dissolve it. Or if not stainless, deal with the staining and etching.
I'm sure there are a few other ways, but that's all that come to mind.
Is this lathe or mill?