Millimeter-wave (mmWave) joint communication-radar (JCR) will enable high data rate communication and high-resolution radar sensing for applications such as autonomous driving. Existing mmWave JCR systems, however, suffer from a limited angular field-of-view and low estimation accuracy for radars due to the use of directional communication beams. In this paper, we propose an adaptive beamforming design for mmWave JCR with a phased-array architecture that permits a trade-off between communication and radar performances. To enable fast estimation of the mmWave radar channel in the Doppler-angle domain, we use a convolutional compressed sensing framework and optimize the radar waveforms within this framework. Our optimization accounts for the space-time sampling constraints that are specific to phased-array radars. We evaluate the JCR performance trade-offs using a normalized mean square error (MSE) metric for radar estimation and a distortion MSE metric for data communication, which is analogous to the distortion metric in the rate-distortion theory. Numerical results demonstrate that our proposed JCR design enables the estimation of short- and medium-range radar channels in the Doppler-angle domain with a low normalized MSE, at the expense of a small degradation in the communication distortion MSE.