Hybrid Analog and Digital (HAD) architectures provide a cost-effective alternative for large-scale antenna arrays, but accurate Direction-of-Arrival (DoA) estimation remains challenging due to limited digital dimensionality and constrained beamforming design. In this work, we propose a HAD architecture that employs Butler matrices to synthesize DFT beams over a uniform linear array. By exploiting the Cauchy-like displacement structure of the beamformed signal, we introduce a second-order statistics estimation algorithm that achieves near-optimal accuracy, approaching the Cramér-Rao Lower Bound (CRLB) and outperforming state-of-the-art methods in simulation.