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I am aware that this is probably a profoundly stupid question. For judges that don't use oscar, how do recommendations work? Do you ask your professor to give you a hard copy that you make a copy of for each individual clerkship application?

It depends on the school. At ours, faculty support and career services coordinate getting letters printed, signed, and then they mail them with the application package you prepare, so you never see the letters (unless recommenders happen to share them with you). Sometimes they will do the mail merge, printing, and signing, and send them to you in sealed envelopes to include with your application. If your school has a clerkship adviser, check what the usual procedure is.