Shared LAMP hosting that supports automation?

Looking suggestions on shared hosts that support some level of automation for website/database setup?

I currently use Dreamhost for several PHP sites. They're fantastic, but they lack the ability to automate the setup of a website. Many of the websites that I set up have the same settings (e.g. include many common PHP scripts and HTML files, include a database, etc..). When I create a website, I need to do it manually through their web control panel. I would like the ability to either create a "template" site that I can copy and use as a starting point, or be able to create a website via an API (so I can do the bootstrapping myself).

I also have several EC2, Linode, Cloudsail, and dedicated servers elsewhere, but I value the fully managed, "shared-hosting" experience that Dreamhost provides for simpler sites that don't need a full node.