The increasing reliance upon cloud services entails more flexible networks that are realized by virtualized network equipment and functions. When such advanced network systems face a massive failure by natural disasters or attacks, the recovery of the entire systems may be conducted in a progressive way due to limited repair resources. The prioritization of network equipment in the recovery phase influences the interim computation and communication capability of systems, since the systems are operated under partial functionality. Hence, finding the best recovery order is a critical problem, which is further complicated by virtualization due to dependency among network nodes and layers. This paper deals with a progressive recovery problem under limited resources in networks with VNFs, where some dependent network layers exist. We prove the NP-hardness of the progressive recovery problem and approach the optimum solution by introducing DeepPR, a progressive recovery technique based on deep reinforcement learning. Our simulation results indicate that DeepPR can obtain 98.4% of the theoretical optimum in certain networks. The results suggest the applicability of Deep RL to more general progressive recovery problems and similar intractable resource allocation problems.