The TESS mission will survey ~85 per cent of the sky, giving us the opportunity of extracting high-precision light curves of millions of stars, including stellar cluster members. In this work, we present our project 'A PSF-based Approach to TESS High quality data Of Stellar clusters' (PATHOS), aimed at searching and characterize candidate exoplanets and variable stars in stellar clusters using our innovative method for the extraction of high-precision light curves of stars located in crowded environments. Our technique of light-curve extraction involves the use of empirical point spread functions (PSFs), an input catalogue and neighbour-subtraction. The PSF-based approach allows us to minimize the dilution effects in crowded environments and to extract high-precision photometry for stars in the faint regime (G>13). For this pilot project, we extracted, corrected, and analysed the light curves of 16641 stars located in a dense region centred on the globular cluster 47 Tuc. We were able to reach the TESS magnitude T~16.5 with a photometric precision of ~1 per cent on the 6.5-h time-scale; in the bright regime we were able to detect transits with depth of ~34 parts per million. We searched for variables and candidate transiting exoplanets. Our pipeline detected one planetary candidate orbiting a main-sequence star in the Galactic field. We analysed the period-luminosity distribution for red-giant stars of 47 Tuc and the eclipsing binaries in the field. Light curves are uploaded on the Mikulski Archive for Space Telescopes under the project PATHOS.