We present the discovery of two sub-Neptune-size planets orbiting HD 63935, a bright (V=8.6, sun-like (Teff=5560) star. Planet b (TOI509.01) was identified in TESS Sector 7 (at twice the orbital period), and planet c appeared in Keck HIRES and Lick APF radial velocity data. We associate planet c to a previously-unidentified single-transit event in the TESS photometry. Our fit of the photometric and radial velocity data yields robust detections of both planets, with respective periods of 9.0600±0.00064 days and 21.35±0.0248 days, radii of 3.058±0.065 Rearth and 2.55±0.23 Rearth, and masses of 11.8±1.8 Mearth and 14.6±2.5 Mearth. For planet b, we calculate a density of 2.49±0.62 g/cm3, consistent with interiors ranging from rocky to ice-dominated, with a hydrogen envelope of at least 1% of the planet mass. Planet c has a higher density of 5.2±2 g/cm3, consistent with a similar range of interiors and a significantly reduced or even absent H envelope. Planet b resides on the sub-Neptune “occurrence cliff”, a drop in short-period planet occurrence around 3 Rearth, separating the sub-Neptune population from giant planets, and is the most promising atmospheric target in that region of parameter space. Planet c is small for a planet of its mass. Together with a small group of other dense sub-Neptunes (e.g. Kepler-48 c, Kepler-131 b, K2-110 b, and K2-263 b), it appears to reside on a spur that protrudes from the sub-Neptune population in the mass-radius diagram. The two planets are consistent with notably distinct compositions, which combined with their unique positions in mass-radius space makes them intriguing subjects for atmospheric characterization. Planet b opens a so far unexplored region of parameter space comprising small (2.6 Rearth < Rp < 4 Rearth), highly irradiated (100 Fearth < Fp < 1000 Fearth) planets around G-stars, none of which have published masses and JWST SNR proxy values as high as HD 63935 b (~120) or published transmission spectra, emphasizing the new regions of parameter space being made accessible to exploration by the TESS mission in general and this system in particular.