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We describe the Perkins INfrared Exosatellite Survey (PINES), a near-infrared photometric search for short-period transiting planets and moons around a sample of 393 spectroscopically confirmed L- and T-type dwarfs. PINES is performed with Boston University's 1.8 m Perkins Telescope Observatory, located on Anderson Mesa, Arizona. We discuss the observational strategy of the survey, which was designed to optimize the number of expected transit detections, and describe custom automated observing procedures for performing PINES observations. We detail the steps of the $\texttt{PINES Analysis Toolkit}$ ($\texttt{PAT}$), software that is used to create light curves from PINES images. We assess the impact of second-order extinction due to changing precipitable water vapor on our observations and find that the magnitude of this effect is minimized in Mauna Kea Observatories $\textit{J}$-band. We demonstrate the validity of $\texttt{PAT}$ through the recovery of a transit of WASP-2 b and known variable brown dwarfs, and use it to identify a new variable L/T transition object: the T2 dwarf WISE J045746.08-020719.2. We report on the measured photometric precision of the survey and use it to estimate our transit detection sensitivity. We find that for our median brightness targets, assuming contributions from white noise only, we are sensitive to the detection of 2.5 $R_\oplus$ planets and larger. PINES will test whether the increase in sub-Neptune-sized planet occurrence with decreasing host mass continues into the L and T dwarf regime.