The Szechuan sauce, which was only made in 1998 to promote the film Mulan, has become well known after featuring in the popular cartoon Rick and Morty.
In July the chain sent a large bottle of the sauce to the show's creators.
They then announced it would come back for one day on 7 October, but fans were left disappointed after stores ran out.
In one video, police can be seen holding back people chanting "we want sauce".
Officers also attended other stores because hundreds of people turned up.