A novel framework to enhance the angular resolution of automotive radars is proposed. An approach to enlarge the antenna aperture using artificial neural networks is developed using a self-supervised learning scheme. Data from a high angular resolution radar, i.e., a radar with a large antenna aperture, is used to train a deep neural network to extrapolate the antenna element's response. Afterward, the trained network is used to enhance the angular resolution of compact, low-cost radars. One million scenarios are simulated in a Monte-Carlo fashion, varying the number of targets, their Radar Cross Section (RCS), and location to evaluate the method's performance. Finally, the method is tested in real automotive data collected outdoors with a commercial radar system. A significant increase in the ability to resolve targets is demonstrated, which can translate to more accurate and faster responses from the planning and decision making system of the vehicle.