An Asymmetric Eclipse Seen toward the Pre-main-sequence Binary System V928 Tau

K2 observations of the weak-lined T Tauri binary V928 Tau A and B show the detection of a single, asymmetric eclipse, which may be due to a previously unknown substellar companion eclipsing one component of the binary with an orbital period >66 days. Over an interval of about 9 hr, one component of the binary dims by around 60%, returning to its normal brightness about 5 hr later. From modeling of the eclipse shape, we find evidence that the eclipsing companion may be surrounded by a disk or a vast ring system. The modeled disk has a radius of 0.9923 ± 0.0005 R<SUB>*</SUB>, with an inclination of 56°78 ± 0°03, a tilt of 41°22 ± 0°05, an impact parameter of -0.2506 ± 0.0002 R<SUB>*</SUB>, and an opacity of 1.00. The occulting disk must also move at a transverse velocity of 6.637 ± 0.002 R<SUB>*</SUB> day<SUP>-1</SUP>, which, depending on whether it orbits V928 Tau A or B, corresponds to approximately 73.53 or 69.26 km s<SUP>-1</SUP>. A search in ground-based archival data reveals additional dimming events, some of which suggest periodicity, but no unambiguous period associated with the eclipse observed by K2. We present a new epoch of astrometry that is used to further refine the orbit of the binary, presenting a new lower bound of 67 yr, and constraints on the possible orbital periods of the eclipsing companion. The binary is also separated by 18″ (˜2250 au) from the lower-mass CFHT-BD-Tau 7, which is likely associated with V928 Tau A and B. We also present new high-dispersion optical spectroscopy that we use to characterize the unresolved stellar binary.