This paper proposes a deep learning approach to the adaptive and sequential beamforming design problem for the initial access phase in a mmWave environment with a single-path channel. For a single-user scenario where the problem is equivalent to designing the sequence of sensing beamformers to learn the angle of arrival (AoA) of the dominant path, we propose a novel deep neural network (DNN) that designs the adaptive sensing vectors sequentially based on the available information so far at the base station (BS). By recognizing that the AoA posterior distribution is a sufficient statistic for solving the initial access problem, we use the posterior distribution as the input to the proposed DNN for designing the adaptive sensing strategy. However, computing the posterior distribution can be computationally challenging when the channel fading coefficient is unknown. To address this issue, this paper proposes to use an estimate of the fading coefficient to compute an approximation of the posterior distribution. Further, this paper shows that the proposed DNN can deal with practical beamforming constraints such as the constant modulus constraint. Numerical results demonstrate that compared to the existing adaptive and non-adaptive beamforming schemes, the proposed DNN-based adaptive sensing strategy achieves a significantly better AoA acquisition performance.