Observations of the emission from spatially extended cold gas around bright high-redshift QSOs reveal surprisingly large velocity widths exceeding 2000 km s^(-1), out to projected distances as large as 30 kpc. The high velocity widths have been interpreted as the signature of powerful AGN-driven outflows. Naively, these findings appear in tension with hydrodynamic models in which AGN-driven outflows are energy-driven and thus very hot with typical temperatures T = 10^6-7 K. Using the moving-mesh code Arepo, we perform 'zoom-in' cosmological simulations of a z = 6 QSO and its environment, following black hole growth and feedback via energy-driven outflows. In the simulations, the QSO host galaxy is surrounded by a clumpy circum-galactic medium pre-enriched with metals due to supernovae-driven galactic outflows. As a result, part of the AGN-driven hot outflowing gas can cool radiatively, leading to large amounts (> 10^9 M_sun) of cold gas comoving with the hot bipolar outflow. This results in velocity widths of spatially extended cold gas similar to those observed. We caution, however, that gas inflows, random motions in the deep potential well of the QSO host galaxy and cooling of supernovae-driven winds contribute significantly to the large velocity width of the cold gas in the simulations, complicating the interpretation of observational data.