Searching for exosatellites orbiting L and T dwarfs: connecting planet formation to moon formation and finding new temperate worlds

L-type and T-type dwarfs span the boundaries between main-sequence stars, brown dwarfs, and planetary-mass objects. For these reasons, L and T dwarfs are the perfect laboratories for exploring the relationship between planet formation and moon formation, and evidence suggests they may be swarming with close-in rocky satellites, though none have been found to date. The discovery of satellites orbiting L or T dwarfs will have transformative implications for the nature of planets, moons and even life in the Universe. These transiting satellites will be prime targets for characterization with NASA's James Webb Space Telescope. In this white paper, we discuss the scientific motivations behind searching for transiting satellites orbiting L and T dwarfs and argue that robotizing current 1-to-2-meter US optical/infrared (O/IR) facilities and equipping them with recently developed low-cost infrared imagers will enable these discoveries in the next decade. Furthermore, robotizing the 1-to-2-meter O/IR fleet is highly synergistic with rapid follow-up of transient and multi-messenger events.