Most generic object detectors are mainly built for standard object detection tasks such as COCO and PASCAL VOC. They might not work well and/or efficiently on tasks of other domains consisting of images that are visually different from standard datasets. To this end, many advances have been focused on adapting a general-purposed object detector with limited domain-specific designs. However, designing a successful task-specific detector requires extraneous manual experiments and parameter tuning through trial and error. In this paper, we first propose and examine a fully-automatic pipeline to design a fully-specialized detector (FSD) which mainly incorporates a neural-architectural-searched model by exploring ideal network structures over the backbone and task-specific head. On the DeepLesion dataset, extensive results show that FSD can achieve 3.1 mAP gain while using approximately 40% fewer parameters on binary lesion detection task and improved the mAP by around 10% on multi-type lesion detection task via our region-aware graph modeling compared with existing general-purposed medical lesion detection networks.