We present observations of two bright M dwarfs (TOI-1634 and TOI-1685: J = 9.5-9.6) hosting ultra-short-period (USP) planets identified by the TESS mission. The two stars are similar in temperature, mass, and radius (Teff ≍ 3500 K, M⋆ ≍ 0.45-0.46 M⊙, and R⋆ ≍ 0.45-0.46 R⊙), and the planets are both super-Earth size (1.25 R⊕ < Rp < 2.0 R⊕). For both systems, light curves from ground-based photometry exhibit planetary transits, whose depths are consistent with those from the TESS photometry. We also refine the transit ephemerides based on the ground-based photometry, finding the orbital periods of P = 0.9893436 ± 0.0000020 days and P = 0.6691416 ± 0.0000019 days for TOI-1634b and TOI-1685b, respectively. Through intensive radial velocity (RV) observations using the InfraRed Doppler (IRD) instrument on the Subaru 8.2 m telescope, we confirm the planetary nature of the TOIs and measure their masses: 10.14 ± 0.95 M⊕ and 3.43 ± 0.93 M⊕ 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 Rp = 1.749 ± 0.079 R⊕ (TOI-1634b) and 1.459 ± 0.065 R⊕ (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 discussing the size limit of rocky planet cores as well as 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. * Based on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.