In this paper, we present the Heavy Metal Survey, which obtained ultra-deep medium-resolution spectra of 21 massive quiescent galaxies at 1.4≲ z≲ 2.2 with Keck/LRIS and MOSFIRE. With integration times of up to 16 hrs 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 (σ_v), ages, and elemental abundances, as detailed in an accompanying paper. Except for one emission-line AGN, all galaxies are confirmed as quiescent through their faint or absent Hα emission and evolved stellar spectra. For most galaxies exhibiting faint Hα, elevated [NII]/Hα suggests a non-star-forming origin. We calculate dynamical masses (M_dyn) by combining σ_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_dyn/M_ and σ_v. This correlation may suggest that a varying IMF, with high-σ_v galaxies being more bottom-heavy, was already in place at z∼2. When implementing the σ_v-dependent IMF found in the cores of nearby early-type galaxies and correcting for biases in our stellar mass and size measurements, we find a low scatter in M_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.