  We present observations of two bright M dwarfs (TOI-1634 and TOI-1685:
$J=9.5-9.6$) hosting ultra-short period (USP) planet candidates, identified by
the TESS mission. The two stars are similar in temperature, mass, and radius,
and the planet candidates are both super-Earth-sized
($1.25\,R_\oplus<R_p<2.0\,R_\oplus$). For both systems, light curves from the
ground-based photometry exhibit planetary transits, whose depths are consistent
with those by the TESS photometry. We also refine the transit ephemerides based
on the ground-based photometry, finding the orbital periods of $P=0.9893457 \pm
0.0000034$ day and $P=0.6691425 \pm 0.0000037$ day for TOI-1634b and TOI-1685b,
respectively. Through intensive radial velocity (RV) observations using the IRD
spectrograph on the Subaru 8.2m telescope, we confirm the planetary nature of
the TOIs, and measure their masses: $9.94\pm0.94\,M_\oplus$ and
$3.84\pm0.96\,M_\oplus$ for TOI-1634b and TOI-1685b, respectively, when the
observed RVs are fitted with a single-planet circular-orbit model. Combining
those with the planet radii of $R_p=1.773\pm 0.077\,R_\oplus$ (TOI-1634b) and
$1.463\pm0.065\,R_\oplus$ (TOI-1685b), we find that both USP planets have mean
densities consistent with an Earth-like internal composition, which is typical
for small USP planets. TOI-1634b is currently the most massive USP planet in
this category, and it resides near the radius valley, which makes it a
benchmark planet in the context of testing the formation scenarios for USP
planets. Excess scatter in the RV residuals for TOI-1685 suggests the presence
of a possible secondary planet or unknown activity/instrumental noise in the RV
data, but further observations are required to check those possibilities.
