One story that I just have to recommend is Fox Fire by WoodLily.
The author says, about it:
Edward's version of "Twilight", written without the aid of "Midnight Sun". Will journey into AU. Rated M for language and sexual content. I couldn't better the "First Sight" character development, therefore the action begins as Edward flees to Alaska...http://www.fanfiction.net/s/5661822/1/Fox_Fire
It's really, very well written! The author captures Edward beautifully, giving us a deeper insight into his character. He isn't some "pure-as-the-driven-snowm" Mormonesque character, like S. Meyer wrote him. Instead, he's a man with the body of a 17 year old, who has an unheatly infatuation with someone he shouldn't care for.
Read it. It's great!!