Automotive radars have an important role in autonomous driving systems. The main challenge in automotive radar detection is the radar's wide point spread function (PSF) in the angular domain that causes blurriness and clutter in the radar image. Numerous studies suggest employing an 'end-to-end' learning strategy using a Deep Neural Network (DNN) to directly detect objects from radar images. This approach implicitly addresses the PSF's impact on objects of interest. In this paper, we propose an alternative approach, which we term "Boosting Radar Reflections" (BoostRad). In BoostRad, a first DNN is trained to narrow the PSF for all the reflection points in the scene. The output of the first DNN is a boosted reflection image with higher resolution and reduced clutter, resulting in a sharper and cleaner image. Subsequently, a second DNN is employed to detect objects within the boosted reflection image. We develop a novel method for training the boosting DNN that incorporates domain knowledge of radar's PSF characteristics. BoostRad's performance is evaluated using the RADDet and CARRADA datasets, revealing its superiority over reference methods.