  We recommend a deeper extension to the High-Latitute Wide Area Survey planned
to be conducted by the Nancy Grace Roman Space Telescope (\emph{Roman}). While
this deeper-tier survey extension can support a range of astrophysical
investigations, it is particularly well suited to characterize the dark matter
substructure in galactic halos and reveal the microphysics of dark matter
through gravitational lensing. We quantify the expected yield of \emph{Roman}
for finding galaxy-galaxy-type gravitational lenses and motivate observational
choices to optimize the \emph{Roman} core community surveys for studying dark
matter substructure. In the proposed survey, we expect to find, on average, one
strong lens with a characterizable substructure per \emph{Roman} tile (0.28
squared degrees), yielding approximately 500 such high-quality lenses. With
such a deeper legacy survey, \emph{Roman} will outperform any current and
planned telescope within the next decade in its potential to characterize the
concentration and abundance of dark matter subhalos in the mass range
10$^7$-10$^{11}$\,M$_{\odot}$.
