I would attempt to grab the adjuster with a Vice-Grip, using some inner tube to protect the part and to provide grip.

Then I would maintain high pulling force while turning the adjuster, hopefully with some of the threads engaging below where the (cast zinc?) threads are probably stripped off of the adjuster.

If this works, possibly some limited amount of adjuster travel can be exploited engaging a few remaining threads on the adjuster

The adjuster appears to have been screwed all the way in with the soft threads having sheared where they engage the nut, but seems there is some threading below the nut threads still holding the adjuster in.