What is Toji's "accent"?
According to this quite old Ask John column (or the Google cache of it, since the original page seems to have gone offline), Touji's accent in Japanese is Osaka-ben. Eva wiki and TV Tropes corroborate this.

In case the cache of the Ask John page disappears, here's a screenshot of it:



You can see in the second paragraph that Touji is listed as a character who speaks with an Osaka accent.

According to the wiki, the accent Touji speaks with in the English manga was supposed to be a New York/Brooklyn accent, which was also used to translate Osaka's accent in the Azumanga Daioh manga. In Abenobashi Mahou Shoutengai and many other shows, a Texas or Southern accent was used.