Recent advancements in Deep Learning (DL) for Direction of Arrival (DOA) estimation have highlighted its superiority over traditional methods, offering faster inference, enhanced super-resolution, and robust performance in low Signal-to-Noise Ratio (SNR) environments. Despite these advancements, existing research predominantly focuses on multi-snapshot scenarios, a limitation in the context of automotive radar systems which demand high angular resolution and often rely on limited snapshots, sometimes as scarce as a single snapshot. Furthermore, the increasing interest in sparse arrays for automotive radar, owing to their cost-effectiveness and reduced antenna element coupling, presents additional challenges including susceptibility to random sensor failures. This paper introduces a pioneering DL framework featuring a sparse signal augmentation layer, meticulously crafted to bolster single snapshot DOA estimation across diverse sparse array setups and amidst antenna failures. To our best knowledge, this is the first work to tackle this issue. Our approach improves the adaptability of deep learning techniques to overcome the unique difficulties posed by sparse arrays with single snapshot. We conduct thorough evaluations of our network's performance using simulated and real-world data, showcasing the efficacy and real-world viability of our proposed solution. The code and real-world dataset employed in this study are available at https://github.com/ruxinzh/Deep_RSA_DOA.