AOSP provides the Android source files. You have someone knowledgeable enough to compile from these sources and build a custom ROM of Lollipop 5.0.1 for Xiaomi mi3 to use it. Besides there are many proprietary files some of these are device specific which is hard to extract (some hackers may do) and needs to be included in the AOSP build.
There are some unofficial AOSP 5.0 Lollipop custom ROM's available. But, it may be lacking many of the original features in your phone. e.g, NFC support or smart gestures etc. AOSP ROM can be built for Google supported devices like Nexus series or some smartphones from Motorola which has Vanilla Android.