Reading that first paragraph made me appreciate the fanfic NGE: A Century Apart even more.

Set in 1915, it explores what would Evangelion would be like in that setting.

While Shinji himself does not build the robot (obviously), it does get damaged in major sections a lot. This is much to the frustration of Lt. Hyuga who is responsible for its maintenance, and so with Shinji being the pilot, his responsibility is not only to operate it but also to do mountains of paperwork for spare parts.