Baked curry, called Yaki kare in Japanese is a local specialty of Moji ward in Kitakyushu city in Fukuoka prefecture. Baked curry was told to be appeared in the menu of a diner located at a covered market street close to Moji port and Moji-ko station in 1950s.
Curry was brought to Japan in late 19th century and developed as the Japanese version and eventually became a national food for Japanese. On the other hand, a dish called doria appeared in Yokohama around 1930 invented by Swiss origin chef. Doria is a gratin using rice which assumed to be a innovated version of French cuisine.
Then, baked curry is a merger of Japanese curry and the doria. It is a curry topped with cheese and baked in a oven as doria. Therefore baked curry is not just the innovation of curry but also an innovation of gratin.
Baked curry can be eaten outside of Moji but it is not so common food which you can find everywhere in Japan. Therefore if you are in Kyushu and see Yaki Kare on the menu, just go for it.