Network slicing and edge computing are key technologies to enable compute-intensive applications for vertical industries in 5G. We define cellular networks with edge computing capabilities as cellular edge computing. In this paper, we study the cross-domain resource orchestration solution for dynamic network slicing in cellular edge computing. The fundamental research challenge is from the difficulty in modeling the relationship between the slice performance and resources from multiple technical domains across the network with many base stations and distributed edge servers. To address this challenge, we develop a distributed cross-domain resource orchestration (DIRECT) protocol which optimizes the cross-domain resource orchestration while providing the performance and functional isolations among network slices. The main component of DIRECT is a distributed cross-domain resource orchestration algorithm which is designed by integrating the ADMM method and a new learning-assisted optimization approach. The proposed resource orchestration algorithm efficiently orchestrates multi-domain resources without requiring the performance model of the network slices. We develop and implement the DIRECT protocol in a small-scale prototype of cellular edge computing which is designed based on OpenAirInterface LTE and CUDA GPU computing platforms. The performance of DIRECT is validated through both experiments and network simulations.