Ok, so that person is not a physician.
Why would the EAD expire ? Is the underlying application (I485) still intact or is this a J2 related EAD ?
A 'plain vanilla' H1b takes a couple of months to process. However, by paying an extra $1000 you can get premium processing which guarantees a 30 day turn-around.
Some positions (e.g. non-profits and universities) are exempt from the H1b cap. Other than that it is quite a gamble these days to rely on availability of a cap-subjected H1b.
Given the PERM labor certification process, with sufficient lead time it might actually be possible to get a green-card up front to start a new job. It mainly depends on the country that the person is from. If there is already a prospective employer for a job to start next summer, it might be possible to get a GC before september (many 'ifs' involved, this assumes a person from iceland working in maine who is able to travel back and forth for consular interviews).