Single-snapshot signal processing in sparse linear arrays has become increasingly vital, particularly in dynamic environments like automotive radar systems, where only limited snapshots are available. These arrays are often utilized either to cut manufacturing costs or result from unintended antenna failures, leading to challenges such as high sidelobe levels and compromised accuracy in direction-of-arrival (DOA) estimation. Despite deep learning's success in tasks such as DOA estimation, the need for extensive training data to increase target numbers or improve angular resolution poses significant challenges. In response, this paper presents a novel Siamese neural network (SNN) featuring a sparse augmentation layer, which enhances signal feature embedding and DOA estimation accuracy in sparse arrays. We demonstrate the enhanced DOA estimation performance of our approach through detailed feature analysis and performance evaluation. The code for this study is available at https://github.com/ruxinzh/SNNS_SLA.