Unagi, freshwater eel brushed with a sweet soy glaze and grilled, is cooked differently from region to region: split from the back or from the belly, steamed before grilling or not. Try the local version wherever your trip takes you, from crisp Osaka eel to Nagoya's hitsumabushi, eaten three ways from one wooden tub. Many of these restaurants are popular and some close when the eel sells out, so go early.
Eel cooked the eastern Japan way: split from the back, grilled plain, steamed, then grilled again with the glaze, so it is soft enough to cut with chopsticks. The eels are kept in spring water from Mt. Fuji before cooking, which takes away any muddy smell. The shop is in Mishima, under an hour from Tokyo by bullet train, and closes for the day when the eel sells out.
Eel cooked the Osaka way: split from the belly and grilled over charcoal without steaming, so the skin turns crisp and smoky. The house dish is mamushi, with more eel hidden in a layer inside the rice. It is a short walk from Yodoyabashi Station in central Osaka.

Nagoya's hitsumabushi: chopped grilled eel on rice in a wooden tub, which you eat in different ways. Have the first bowl plain, the second with condiments such as scallions and wasabi, and the third with broth poured over like a soup. Lines form at mealtimes, so arrive early; the shop is near Atsuta Shrine.
Yanagawa's seiro-mushi: rice mixed with the sweet glaze, topped with grilled eel and thin strands of egg, then steamed in a bamboo box called a seiro, so every grain soaks up the sauce. The shop says it has served it for more than 340 years. Yanagawa is about 50 minutes from Fukuoka by train, and the shop is closed on Mondays.
Kyoto's kinshi-don: a bowl of eel on rice under a big, soft rolled omelet. Break the egg and the glazed eel appears underneath, so you eat the two together. The shop is near the Shinkyogoku shopping arcade, a short walk from Kawaramachi Station, and is closed on Wednesdays.
A restaurant run by the fish farmers' cooperative of Lake Hamana, where eel farming in Japan began, serving only eel raised around the lake. Order unaju, grilled eel over rice in a lacquered box. It is near Hamamatsu Station, about 1.5 hours from Tokyo by bullet train.
A thatched-roof restaurant in Okaya, a town in Nagano that loves eel, where the eel is grilled slowly over charcoal without steaming until the skin is crisp. Okaya also eats eel on a day in midwinter, so it is worth a visit in the cold season too. It is about 10 minutes by taxi from Okaya Station, and closed on Thursdays.