Jeff Sessions himself picked Rod Rosenstein on the recommendation of Sessions’ long-time pal Chuck Cooper (of Cooper & Kirk). Cooper is considered one of the top few litigators in DC. Rosenstein was also promoted by Chris Christie.
Cooper was with Sessions when Sessions interviewed Rosenstein. And Cooper probably approved of Rosenstein after the interview. But I believe Cooper didn't know Rosenstein before he met him at the interview.
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So I don't think it was Cooper who got Rosenstein the interview. As you say it might have been Christie.