TESSILATOR: a one-stop shop for measuring TESS rotation periods

We present a software package designed to produce photometric lightcurves and measure rotation periods from full-frame images taken by the Transiting Exoplanet Survey Satellite (TESS), which we name ``TESSILATOR''. TESSILATOR is the only publicly-available code that will run a full lightcurve and rotation period ($P_{\rm rot}$) analysis based on just a (list of) target identifier(s) or sky position(s) via a simple command-line prompt. This paper sets out to introduce the rationale for developing TESSILATOR, and then describes the methods, considerations and assumptions for: extracting photometry; dealing with potential contamination; accounting for natural and instrumental systematic effects; lightcurve normalisation and detrending; removing outliers and unreliable data; and finally, measuring the $P_{\rm rot}$ value and several periodogram attributes. Our methods have been tuned specifically to optimise TESS lightcurves and are independent from the pipelines developed by the TESS Science Processing Operations Center, meaning TESSILATOR can, in principle, analyse {\it any} target across the entire celestial sphere. We compare TESSILATOR $P_{\rm rot}$ measurements with TESS-SPOC-derived lightcurves of 1,560 (mainly FGKM-type) stars across four benchmark open clusters (Pisces-Eridanus, the Pleiades, the Hyades and Praesepe) and a sample of nearby field M-dwarfs. From a vetted subsample of 864 targets we find an excellent return of $P_{\rm rot}$ matches for the first 3 open clusters ($>85$ per cent) and a moderate ($\sim 60$ per cent) match for the 700 Myr Praesepe and MEarth sample, which validates TESSILATOR as a tool for measuring $P_{\rm rot}$. The TESSILATOR code is available at \url{https://github.com/alexbinks/tessilator}.