Massive data traffic over capacity-limited fronthaul link of a cloud radio access network (CRAN) poses a severe challenge in modern wireless systems. We propose a novel CRAN architecture to address this challenge that leverages a large array of antennas or massive MIMO at the baseband unit (BBU) to communicate with a set of distributed remote radio heads (RRHs) and subsequently serve multiple user equipment (UEs) over the access link. Transmission of data from massive MIMO at BBU takes place over a millimeter wave (mmWave) channel to exploit large bandwidth. The proposed architecture leverages a substantial array gain of massive MIMO in conjunction with a large idle mmWave frequency spectrum that can deal with a wide range of throughput requirements. In addition, a data compression strategy is designed by finding an optimal quantization noise covariance matrix across all the RRHs to handle the fronthaul traffic load. To this end, an optimization problem to maximize the sum rate of the UEs is thus formulated subject to constraints on fronthaul capacity, data compression, and total transmit powers at the BBU and RRHs. To solve this non-convex and intractable problem, a path following an iterative algorithm based on the difference of convex (DC) programming is used. The performance of the proposed architecture is evaluated in comparison with nontrivial benchmark schemes/architectures. Simulation results reveal that the proposed architecture can significantly improve the throughput of a CRAN by overcoming the fronthaul bottleneck by using massive MIMO over the mmWave channel.