Sponsored H1B vs. green card self-petition: What's the quickest? I am a postdoctoral scholar at a top ranked US University and I am on a J1 visa. I have received an offer for a position in a company, which should be filled by May 2016. The company told me that they may sponsor me for a H1B visa, but the asked me to check whether it may be faster to apply for a green card with a self-petition as an academic of Extraordinary Ability (EBI-1A). Time is what counts most for them now, and they won't be able to hire me if I don't get this done before May- which I think it will be tough.

I would appreciate to receive your advice on this. I moved recently to the US, and I am still misinformed and very naive on this issue.
US immigration processes are a chancy and fast-moving target, as well as being subject to arbitrary interference from the political system.  I have been peripherally involved with an "extraordinary ability" application before, and my understanding from that process is that it's awesome if you can get it, but highly unpredictable.  I would thus strongly recommend against counting on EBI-1A.

On the other hand, H-1B also has its downsides: the number of available H-1B visas is capped through a highly complex formula.  You also can end up "hostage" to a poor employment situation because you go immediately out of status if you quit or are fired.  You also count as a non-US person, subject to export controls and potential limitations on the types of work that you can do---with some companies this is no constraint, with others it severely limits your career.

Instead, the first thing I would recommend is to get a trustworthy immigration lawyer involved to help you through the process.  See if it's possible to do a "belt and suspenders" approach where you simultaneously aim for the extraordinary ability green-card (since getting a green-card gives you more personal and professional flexibility) and the H-1B route with the company for backup.  Even with the company taking charge of an H-1B, it's good to have a US lawyer advocating for you also, because if a random low-level clerical mistake at the company may be "Oops, too bad!" for them but a wrenching restructuring of your life plans for you.