Looking for software so I can spin up a quick test site with temp URL based on GIT feature branch

I'm looking for some software (web based ideally) that will allow me to spin up quick clones of my live website(s) for client approval/testing purposes on to a temporary URL. Ideally I'd be able to specify a GIT feature branch to deploy to it so my client only sees the exact features they are approving (as there could be lots of work in progress on the current test site).

As an example: My client wants to see what a new module looks like on mysite.com. Currently I would add the module to my local project and test (wamp), commit to GIT feature branch, push, then create a new staging/test server by dumping/tarring, uploading, creating hosting space, etc etc etc. This is really time consuming and expensive.

I don't even know if anything like this even exists but if it does I'd love to hear about it! Thanks.