On level 1-3, just put one of your chefs by the wash station and have your other chef start serving dish with nothing on it. It will dirty the plates quickly and the dirty plates gets returned right next to the washing station so you can just continue to clean plates without having to do much running around other than grabbing the clean plates. As with most achievements on here if you have a second player willing to help it will go much faster but doing it solo will also go pretty fast.
Note you have to be the one actually cleaning the dishes for it to count. Also the tracker does not go up until after you have completed the mission.
Edit: per Elite1111111111 you can serve the plate with nothing on it.