The JWST Disk Infrared Spectral Chemistry Survey (JDISCS) aims to understand the evolution of the chemistry of inner protoplanetary disks using the Mid-InfraRed Instrument (MIRI) on the James Webb Space Telescope (JWST). With a growing sample of >30 disks, the survey implements a custom method to calibrate the MIRI Medium Resolution Spectrometer (MRS) to contrasts of better than 1:300 across its 4.9–28 μm spectral range. This is achieved using observations of Themis family asteroids as precise empirical reference sources. The high spectral contrast enables precise retrievals of physical parameters, searches for rare molecular species and isotopologues, and constraints on the inventories of carbon- and nitrogen-bearing species. JDISCS also offers significant improvements to the MRS wavelength and resolving power calibration. We describe the JDISCS calibrated data and demonstrate their quality using observations of the disk around the solar-mass young star FZ Tau. The FZ Tau MIRI spectrum is dominated by strong emission from warm water vapor. We show that the water and CO line emission originates from the disk surface and traces a range of gas temperatures of ∼500–1500 K. We retrieve parameters for the observed CO and H2O lines and show that they are consistent with a radial distribution represented by two temperature components. A high water abundance of n(H2O) ∼ 10−4 fills the disk surface at least out to the 350 K isotherm at 1.5 au. We search the FZ Tau environs for extended emission, detecting a large (radius of ∼300 au) ring of emission from H2 gas surrounding FZ Tau, and discuss its origin.