It can be done
I did it by calling AMEX directly (they asked for the number and I said I didn't have it). There is no way to do it using the website. Also, keep in mind when the new card arrives, you have to activate it over the phone or on the website and it asks for a social security number again.
I went to the website and put in my SSN and it was activated.
Since I am responsible for the account, I think their attempt to collect additional info on anyone I choose to give a card to is ridiculous. I have never been asked to provide similar info from my MC or discover card. Also, I have been a AMEX Platinum Card holder since 1997 with a perfect payment history.