In this paper, a new method of fast beam search and refinement is proposed for millimeter-wave hybrid beamforming systems. The proposed method is based on a two-level phased array approach in which the first-level phased array is formed by actual analog-domain subarrays, and the second-level phased array is virtually formed by the outputs of the first-level subarray phased combiners. Exploiting the fact that the overall beam pattern of the two-level array is the product of the beam patterns of the two levels, the proposed method searches angle-of-arrivals by sweeping coarse-resolution analog training beams at the first-level phased array and matching the first-level subarray outputs with fine-resolution digital training beams by parallel fast Fourier transform (FFT) filtering. In this way, the overall beam search resolution of the proposed method is given by the fine resolution of the second-level virtual array only with the overhead of coarse beam sweeping at the first-level subarrays. Hence, the proposed method provides a very efficient way of beam training and refinement. The performance of the proposed method including the directional ambiguity, beam search latency, and computational complexity is analyzed. Numerical results show the effectiveness of the proposed method.