Third-generation gravitational wave detectors will use large mirrors isolated from seismic motion at low frequency, and also cooled down to cryogenic temperatures. To fulfil these two specifications, the E-TEST project explores the possibility of using a purely non-contact radiative cooling strategy. Based on cooling predictions, the paper includes a detailed design of the cryostat and the assembly procedure. A test campaign demonstrated that the proposed strategy succeeded in bringing the temperature of a [Figure presented] dummy mirror down to [Figure presented] in 19 days. These encouraging results are paving the way toward a fully radiative approach for cooling the mirrors of the future Einstein Telescope.