Meteoritic chondrules were formed in the early solar system by brief heating of silicate dust to melting temperatures. Some highly refractory grains (Type B calcium-aluminum-rich inclusions, CAIs) also show signs of transient heating. A similar process may occur in other protoplanetary disks, as evidenced by observations of spectra characteristic of crystalline silicates. One possible environment for this process is the turbulent magnetohydrodynamic flow thought to drive accretion in these disks. Such flows generally form thin current sheets, which are sites of magnetic reconnection, and dissipate the magnetic fields amplified by a disk dynamo. We suggest that it is possible to heat precursor grains for chondrules and other high-temperature minerals in current sheets that have been concentrated by our recently described short-circuit instability. We extend our work on this process by including the effects of radiative cooling, taking into account the temperature dependence of the opacity; and by examining current sheet geometry in three-dimensional, global models of magnetorotational instability. We find that temperatures above 1600 K can be reached for favorable parameters that match the ideal global models. This mechanism could provide an efficient means of tapping the gravitational potential energy of the protoplanetary disk to heat grains strongly enough to form high-temperature minerals. The volume-filling nature of turbulent magnetic reconnection is compatible with constraints from chondrule-matrix complementarity, chondrule-chondrule complementarity, the occurrence of igneous rims, and compound chondrules. The same short-circuit mechanism may perform other high-temperature mineral processing in protoplanetary disks such as the production of crystalline silicates and CAIs.