I know it's a pain. I think we've all had snap-off experience!
The pick idea has worked for me a few times when I drill the hole to the point that I can see the hole threads. Be careful not to drill the threads out.
If you can pick the thin walled remnant away and grab a hold with some needle-nose you could colapse the remnant more and unscrew it out the rest of the way.
You might hand tap in the hole afterwards to chase the threads.
There are a lot of experienced ideas shared here. I like that!
Jerry
