Just a thought on how the tap may have broken…
…. If you were not using some form of tap-guide in a bench-drill, or similar aid.
If you used the same set-up in the vice for the tapping as illustrating the extracting, the adaptor might have twisted slightly sideways, rotating about the very small contact areas, so suddenly throwing a side force on the tap.
I would suggest either using soft V- or channel- blocks (e.g. cut from thick plywood) between the work and vice-jaws; or clamping the work down to a solid surface – e.g. the bench or a piece of wood in the vice.
With similar annular objects, I've exploited the central hole. I've fitted the item to a bench-drill table-slot, or board with 10mm through-hole and held in the vice or G-clamped to the bench; using a stud, cross-bar and nuts (e.g. from the milling-machine clamp-set).
Another point I have found is work-height relative to own height – not much in my case. It feels to me that hand tap-wrench or die-holder handles above elbow-height make it harder to produce even, balanced forces on the tool.