We present an architecture for the provision of video Content Delivery Network (CDN) functionality as a service over a multi-domain cloud. We introduce the concept of a CDN slice, that is, a CDN service instance which is created upon a content provider's request, is autonomously managed, and spans multiple potentially heterogeneous edge cloud infrastructures. Our design is tailored to a 5G mobile network context, building on its inherent programmability, management flexibility, and the availability of cloud resources at the mobile edge level, thus close to end users. We exploit Network Functions Virtualization (NFV) and Multi-access Edge Computing (MEC) technologies, proposing a system which is aligned with the recent NFV and MEC standards. To deliver a Quality-of-Experience (QoE) optimized video service, we derive empirical models of video QoE as a function of service workload, which, coupled with multi-level service monitoring, drive our slice resource allocation and elastic management mechanisms. These management schemes feature autonomic compute resource scaling, and on-the-fly transcoding to adapt video bit-rate to the current network conditions. Their effectiveness is demonstrated via testbed experiments.