Low Gain Avalanche Diodes (LGADs) are thin (20-50 $mu m$)silicon di ode sensors with modest internal gain (typically 5 to 50) and exceptional time resolution (17 $ps$ to 50 $ps$). However, the granularity of such devices is limited to the millimeter scale due to the need to include protection structures at the boundaries of the readout pads to avoid premature breakdown due to large local electric fields. In this paper we present a new approach -- the Deep-Junction LGAD (DJ-LGAD) -- that decouples the high-field gain region from the readout plane. This approach is expected to improve the achievable LGAD granularity to the tens-of-micron scale while maintaining direct charge collection on the segmented electrodes.