So what do I do? I tear open the wall to install some "nailers" that will accept lag screws.
This wall is right against the chimney, but even so, what's up with the 20" on center stud bay? I decided not to block in the big bay for drywall purposes, but I'm sure that I should have. All I wanted was a sink I could lean on!
Too bad I'm a terrible taper. Also too bad to not have a bathroom sink for a minute....