The Deep Underground Neutrino Experiment (DUNE) will employ a uniquely large Liquid Argon Time Projection Chamber with four separate modules of 10kt fiducial mass each. Different modules will utilize single and dual-phase Liquid Argon technologies. An experimental program ("protoDUNE") has been initiated which includes a beam and cosmic ray test of large-scale DUNE prototypes at CERN in 2018. The volume of data to be collected by the protoDUNE single-phase detector will amount to a few petabytes and the sustained rate of data sent to mass storage will be in the range of a few hundred megabytes per second. The protoDUNE experiment requires substantial Data Quality Monitoring capabilities in order to ascertain the condition of the detector and its various subsystems. To this end, a Prompt Processing system has been developed which is complementary to Online Monitoring and is characterized by a lower bandwidth, scalable CPU resources and end-to-end latency on the scale of a few minutes. We present the design of the protoDUNE Prompt Processing system, the current status of its development and testing and issues related to its interfaces and deployment.