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In quiescence, Sgr A* is surprisingly dim, shining 100,000 times less than expected for its environment. This problem has motivated a host of theoretical models to explain radiatively inefficient accretion flows (RIAFs). The Chandra Galactic Center (GC) X-ray Visionary Program obtained approximately 3 Ms (one month) of Chandra HETG data, offering the only opportunity to examine the quiescent X-ray emission of Sgr A* with high resolution spectroscopy. Utilizing custom background regions and filters for removing overlapping point sources, this work provides the first ever look at stacked HETG spectra of Sgr A*. We model the background datasets with a cubic spline and fit the unbinned Sgr A* spectra with a simple parametric model of a power law plus Gaussian lines under the effects of interstellar extinction. We detect a strong 6.7 keV iron emission line in the HEG spectra and a 3.1 keV emission line in the MEG spectra. In all cases, the line centroids and equivalent widths are consistent with those measured from low-resolution CCD spectra. An examination of the unbinned, stacked HEG+/-1 spectrum reveals fine structure in the iron line complex. In addition to resolving the resonant and forbidden lines from He-like iron, there are apparent emission features arising with higher statistical significance at lower energy, potentially associated with FeXX-XXIV ions in a ~1 keV plasma arising near the Bondi radius of Sgr A*. With this work, we release the cleaned and stacked Sgr A* and background HETG spectra to the public as a special legacy dataset.