Insist that candidates submit dossiers that contains blinded copies of everything, such that the files that are prepared by the staff and then passed on to the hiring committee members are assigned only a number, containing no information as to the candidate’s name, gender, or associated institutions.Of course, as many are pointing out in the comments over yonder, there are practical issues with having blind dossiers, but there's also those arguing against blind letters more theoretically.
These arguments emphasize that blinding cutting out valuable information or makes it difficult to "calibrate the praise" in letters. But, on what scale are we doing the calibration and what valuable information is being left off? Uncharitably, the answer is obvious: the scale and information being left off is pedigree and ranking. I'd be more willing to be less uncharitable if there was any indication that something besides this information would be missing in blind dossiers.
But, in the end, maybe the arguments are good. Isn't it just obvious that the best student in 20 years at Pittsburgh is better than the best student in 20 years at Washington University? I mean, come on!