Then I have a funny feeling you'll love this: http://www.fanfiction.net/s/7366113/1/Shinji-and-the-Doctor

It's ongoing, and updates seem to be fairly frequent.

Basically, the (10th) Doctor crashes into the Evangelion-verse, meets Shinji, deals with a zombie, figures out there's something weird about the whole place and then does his usual thing of running away and forgetting about it trying to save the world in his usual Crazy Awesome style.