We report on observations of the soft X-ray transient (SXT) GRO J0422+32 made with the ROSAT Position Sensitive Proportional Counter (PSPC) and the ASCA GIS and SIS X-ray imaging spectrometers during its 1993 August outburst. This event occurred approximately one year after the primary outburst of the source and was characterized by an X-ray-to-optical luminosity ratio much smaller than that typically seen in SXT outbursts or in steady, low-mass X-ray binaries. Our X-ray observations with the ROSAT PSPC, spread over 6.2 days and totalling 28.8 ks, combined with our archival analysis of 17 ks of ASCA observations, indicate an approximately 1 mcrab source that turned and subsequently back on over a 10 day period, with distinct spectral characteristics in each case. In addition to our analysis of the X-ray spectra, we report on optical photometry and spectroscopy bracketing the X-ray observations. It is noteworthy that the optical light curves remain approximately constant, at least in our data's almost daily sampling, during the turning and turning on sequence seen in the X-rays. Optical spectroscopy reveals large, asymmetric redshifted and blueshifted components in the double-peaked emission lines He II λ4686 and Hβ λ4861, which can be interpreted as evidence for a wind emanating from the secondary. The phasing of these shifts ruled out a wind originating from the disk. On the other hand, the X-ray power illuminating the star seems insufficient for driving such a wind. Comparisons are made with the primary (1992) outburst and decay phase of GRO J0422+32. Implications for X-ray reprocessing scenarios of the production of optical and UV emission and for other possible scenarios, such as advection-dominated accretion, are briefly considered. An upper limit to the quiescent X-ray luminosity is estimated using data acquired during the X-ray off state. A similarity between the X-ray-to-optical behavior of GRO J0422+32 during this minioutburst and previously documented optical-to-EUV behavior of certain SU Ursae Majoris system outbursts is noted. A physical interpretation of this analogous behavior, suggesting that the hot and cold branches of the S-shaped stability curve may be relatively closely spaced in GRO J0422+32, is offered.