Closest Thing To AOSP? (5.1.1 or 6.0)

Hi there,
I've done a bit of searching, and not found much, so I thought I would ask. What ROM is the closest I can get to having fully functional AOSP on my device, and how would I go about rooting it?

I have built Android 5.1.1 and Android 6 for my device as per Sony's instructions but I hear 6.0 is not stable enough, and I'm not sure about how to go about rooting 5.1.1.

Thanks for any help you can provide!