Context. Gas giant planets orbiting low-mass stars (Teff ≲ 4600 K) are uncommon outcomes of planet formation. Increasing the sample of well-characterised giants around early M dwarfs will enable population-level studies of their properties, offering valuable insights into their formation and evolutionary histories. Aims. We aim to confirm and characterise giant exoplanets transiting M dwarfs identified by the TESS mission. To this end, we have started the Gas giAnts Transiting 1Ow-mass Stars (GATOS) programme within the NIRPS guaranteed time observations (GTO). Methods. High-resolution spectroscopic data were obtained in the optical and near-infrared (nIR), combining HARPS and NIRPS. We derived radial velocities (RVs) via the cross-correlation function and implemented a novel post-processing procedure to further mitigate telluric contamination in the nIR. The resulting RVs were jointly fit with TESS and ground-based photometry to derive the orbital and physical parameters of the systems. Results. We present the GATOS programme and its first results. We confirm two gas giants transiting the low-mass stars TOI-3288 A (K9V, Teff = 3933 ± 48 K) and TOI-4666 (M2.5V, Teff = 3512 ± 36 K). TOI-3288 A hosts a hot Jupiter with a mass of 2.11 ± 0.08 MJup and a radius of 1.00 ± 0.03 RJup, with an orbital period of 1.43 days (Teq = 1059 ± 20 K). TOI-4666 hosts a 0.70 ± 0.06 MJup warm Jupiter (Teq = 713 ± 14 K) with a radius of 1.11 ± 0.04 RJup, with an orbital period of 2.91 days. At a population level, we identify a decrease in planetary mass with spectral type, whereby late M dwarfs host less massive giant planets than early M dwarfs. More massive gas giants that deviate from this trend are preferentially hosted by more metal-rich stars. Furthermore, we find an increased binarity fraction among low-mass stars hosting gas giants, which may play a role in enhancing giant planet formation around low-mass stars. Conclusions. These mass characterisations contribute to the growing catalogue of well-defined giant exoplanets around low-mass stars. The observed population trends agree with theoretical predictions, whereby higher metallicity can compensate for lower disc masses, and wide binary systems may influence planet formation and migration through Kozai─Lidov cycles or disc instabilities.