Millimeter-wave (mmWave) joint communicationradar (JCR) will enable high data rate communication and highresolution radar sensing for applications such as autonomous driving. Prior JCR systems that are based on the mmWave communications hardware, however, suffer from a limited angular field-of-view and low estimation accuracy for radars due to the employed directional communication beam. In this paper, we propose an adaptive and fast combined waveform-beamforming design for the mmWave automotive JCR with a phased-array architecture that permits a trade-off between communication and radar performances. To rapidly estimate the mmWave automotive radar channel in the Doppler-angle domain with a wide field-of-view, our JCR design employs circulant shifts of the transmit beamformer to acquire radar channel measurements and uses two-dimensional compressed sensing (CS) in the spacetime dimension. We optimize these circulant shifts to minimize the coherence of the CS matrix, under the space-time sampling constraints in our problem. 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. Additionally, we develop a MSEbased weighted average optimization problem for the adaptive JCR combined waveform-beamforming design. 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.