Hi,
I had similar situation, and I suggest you to contact my lawyer for a legal letter to accompany the evidence you are going to send them.
The "problem" is:
The lapse between your OPT and the end-date on your I-20. Actually this is not illegal, because State of Department specifies that there is 60-day grace period for F-1 student to leave the country after completetion of academic study. However this is grey area in law. If you want to contact my lawyer, here is his contacts: Howard Lee, 212-689-0700, he is very good and responsible.
If you don't want to contact him, then included in your evidence package, send (1) the page from the State of Department about F-1 status visa grace period, (2) a copy of your original application package for EAD, (this would show that you intended to apply EAD before your I20 end date, it was INS that issued it late); (3) cite of 245(i)(K) if INS law, to say that you are entitled to adjust your status even if for argument purpose, that one month lag was not deemed legal, because of 180-day "out-of-status" limit (see the 245i(k)itself, do use my language), etc.
Good Luck. Your evidence and legal letter would be good enough, but be careful, because Intent of Deny is pretty serious (I didn't get such letter, but had an interview).