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Media streaming is the dominant application over wireless edge (access) networks. The increasing softwarization of such networks has led to efforts at intelligent control, wherein application-specific actions may be dynamically taken to enhance the user experience. The goal of this work is to develop and demonstrate learning-based policies for optimal decision making to determine which clients to dynamically prioritize in a video streaming setting. We formulate the policy design question as a constrained Markov decision problem (CMDP), and observe that by using a Lagrangian relaxation we can decompose it into single-client problems. Further, the optimal policy takes a threshold form in the video buffer length, which enables us to design an efficient constrained reinforcement learning (CRL) algorithm to learn it. Specifically, we show that a natural policy gradient (NPG) based algorithm that is derived using the structure of our problem converges to the globally optimal policy. We then develop a simulation environment for training, and a real-world intelligent controller attached to a WiFi access point for evaluation. We empirically show that the structured learning approach enables fast learning. Furthermore, such a structured policy can be easily deployed due to low computational complexity, leading to policy execution taking only about 15s. Using YouTube streaming experiments in a resource constrained scenario, we demonstrate that the CRL approach can increase QoE by over 30%.