In this paper, we present an application-specific Multi-Access Edge Computing (MEC) network architecture by leveraging the Control and User Plane Separation (CUPS) in mobile core networks to offload data processing from central servers to edge servers to reduce the transmitted traffic volume and also the response latency of connected vehicle mobility service. We first apply deep learning to classify packets of different applications to different Radio Access Networks (RAN) slices for application-specific spectrum scheduling. Then, we slice Evolved Packet Core (EPC) and deploy EPC data plane slices on-demand for each application and route packets from RAN slices to edge servers. By applying network slicing, multiple RAN, EPC and MEC slices that support different categories of services with different quality of service (QoS) requirements can be deployed in the same physical infrastructure. We prototype the proposed application-specific CUPS architecture using modified open source software OpenAirInterface on our deeply programmable platform. The preliminary experimental results show the feasibility and efficiency of proposed application-specific CUPS architecture, which can achieve a significant decrease in transmission data volume and latency.