We report the discovery of NGTS-11 b (=TOI-1847 b), a transiting Saturn in a 35.46-day orbit around a mid K-type star (Teff=5050 K). We initially identified the system from a single-transit event in a TESS full-frame image light-curve. Following seventy-nine nights of photometric monitoring with an NGTS telescope, we observed a second full transit of NGTS-11 b approximately one year after the TESS single-transit event. The NGTS transit confirmed the parameters of the transit signal and restricted the orbital period to a set of 13 discrete periods. We combined our transit detections with precise radial velocity measurements to determine the true orbital period and measure the mass of the planet. We find NGTS-11 b has a radius of 0.817+0.028-0.032 $RJ$, a mass of 0.344+0.092-0.073 $MJ$, and an equilibrium temperature of just 435+34-32 K, making it one of the coolest known transiting gas giants. NGTS-11 b is the first exoplanet to be discovered after being initially identified as a TESS single-transit event, and its discovery highlights the power of intense photometric monitoring in recovering longer-period transiting exoplanets from single-transit events.