Stuck brake adjuster advice sought

[bikeworkshop]

Because its a barrel adjustor it already has a central hole in it -
I'd suggest drilling slowly from the underside with a 4mm bit. The chances are that as the drill bites, the adjuster will simply unscrew.

A bolt extractor would get it out from the other side - they are quite cheap.

 

[Woz]

Would at least start swearing at it. If you've decided the drill route is good, beforehand at least get medieval on it's ass.