Utsunomiya, a city north of Tokyo is famous for Gyoza and just southwest of the main exit of the station (near bus stop 6) is my favorite gyoza shop. I'm reluctant to say it's the best because I have only been to a few of the 100's of shops in Utsunomiya.
If you happen to be in Japan and near Tokyo or Ueno station with a JR rail pass, it's really easy to hop on a bullet train and within 45 minutes be standing at the door of this shop. The menu is below, like most shops, it's simple with about 4 or 5 items. It's not really important to be able to read this too, because the people around you have all ready ordered everything on the menu and they can help you.
He is preparing the stove for the next set of gyoza. I'm not sure what the name of this shop is.
(Update: June 2012)
Another gyoza recommendation is a small shop called Masashi, about a 15 minute walk from Utsunomiya station.
They only serve two things, fired gyoza and soup gyoza... no beer! Above is two orders of fried gyoza, cost is 440yen. It can be hard to fine, here is a link to google map.