Copper is diamond-turnable.
Look into diamond turning. One could possibly achieve this, using a new, round diamond tool and by taking a series of very light passes at different pass angles, but doubt this would be practical. It will be important to know if the copper object is cylindrical, or flat, as the tooling would differ. Either form, however, should be capable of being smoothed to this finish by burnishing with either a hardened, polished wheel on a lathe, or passing between rolls, ground and polished to at least this level of smoothness. Some metals suppliers are equipped to roll copper to mirror-like finishes, then interleave a special paper to protect the surface for shipping, as another possibility.