We report the discovery of NGTS-11 b (=TOI-1847b), a transiting Saturn in a 35.46 day orbit around a mid K-type star ( = 5050 80 K). We initially identified the system from a single-transit event in a TESS full-frame image light curve. Following 79 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 , a mass of , and an equilibrium temperature of just 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.