The directionality of millimeter-wave (mmWave) communications creates a significant challenge in serving fast-moving mobile terminals on, e.g., high-speed vehicles, trains, and UAVs. This challenge is exacerbated in mmWave systems using analog antenna arrays, because of the inherent non-convexity in the control of the phase shifters. In this paper, we develop a recursive beam tracking algorithm which can simultaneously achieve fast tracking speed, high tracking accuracy, low complexity, and low pilot overhead. In static scenarios, this algorithm converges to the minimum Cramér-Rao lower bound (CRLB) of beam tracking with high probability. In dynamic scenarios, even at SNRs as low as 0dB, our algorithm is capable of tracking a mobile moving randomly at an absolute angular velocity of 10-20 degrees per second, using only 5 pilot symbols per second. If combining with a simple TDMA pilot pattern, this algorithm can track hundreds of high-speed mobiles in 5G configurations. Our simulations show that the tracking performance of this algorithm is much better than several state-of-the-art algorithms.