  In this paper, we present the Heavy Metal Survey, which obtained ultradeep
medium-resolution spectra of 21 massive quiescent galaxies at $1.3<z<2.3$ with
Keck/LRIS and MOSFIRE. With integration times of up to 16\,hr per band per
galaxy, we observe numerous Balmer and metal absorption lines in atmospheric
windows. We successfully derive spectroscopic redshifts for all 21 galaxies and
for 19 we also measure stellar velocity dispersions ($\sigma_v$), ages, and
elemental abundances, as detailed in an accompanying paper. Except for one
emission-line active galactic nucleus, all galaxies are confirmed as quiescent
through their faint or absent H$\alpha$ emission and evolved stellar spectra.
For most galaxies exhibiting faint H$\alpha$, elevated [NII]/H$\alpha$ suggests
a non-star-forming origin. We calculate dynamical masses ($M_{\rm dyn}$) by
combining $\sigma_v$ with structural parameters obtained from
HST/COSMOS(-DASH), and compare them with stellar masses ($M_*$) derived using
spectrophotometric modeling, considering various assumptions. For a fixed
initial mass function (IMF), we observe a strong correlation between $M_{\rm
dyn}/M_*$ and $\sigma_v$. This correlation may suggest that a varying IMF, with
high-$\sigma_v$ galaxies being more bottom heavy, was already in place at
$z\sim2$. When implementing the $\sigma_v$-dependent IMF found in the cores of
nearby early-type galaxies \textit{and} correcting for biases in our stellar
mass and size measurements, we find a low scatter in $M_{\rm dyn}/M_*$ of 0.14
dex. However, these assumptions result in unphysical stellar masses, which
exceed the dynamical masses by 34%. This tension suggests that distant
quiescent galaxies do not simply grow inside-out into today's massive
early-type galaxies and the evolution is more complicated.
