Rather than try and reconfigure the Amazon Linux AMI to use Nginx and PHP-FPM, it was much easier to reconfigure Apache to use the Event MPM instead of Prefork, and utilize PHP-FPM. This uses the 64bit Amazon Linux 2014.09 v1.2.0 running PHP 5.5. Save this file in your .ebextensions folder in the root of your project.
This utilized mod_proxy_handler from Arch Linux. For mod_proxy_handler.c, please see the link in that “file” section below. Update: This no longer uses the mod_proxy_handler module for Apache, as (a) Arch Linux removed the download and (b) support for what it did is included in Apache 2.4.10, which is what is included in the AMI this now works with.