An illustrative media agent (MA) in a data storage management system instructs a NAS file server (filer) to restore an MA-created synthesized-copy instead of larger filer-created backup copies. The synthesized-copy is designed only for the particular files to be restored and mimics, and is typically much smaller than, a filer-created backup copy. The synthesized-copy is fed to the filer on restore as a “restore data image.” When receiving a restore request for certain backed-up data files, the MA synthesizes the synthesized-copy on the fly. The MA generates a header mimicking a filer-created backup header; extracts files from filer-created backup copies arranging them within the synthesized-copy as if in filer-created backups; and instructs filer to perform a full-volume restore from the synthesized-copy. The MA serves the synthesized-copy piecemeal as available, rather than waiting to synthesize the entire synthesized-copy. The synthesized-copy is not stored at the MA.