The detection of the unusually heavy binary neutron star merger GW190425 marked a stark contrast to the mass distribution from known Galactic pulsars in double neutron star binaries and gravitational-wave source GW170817. We suggest here a formation channel for heavy binary neutron stars and light black hole-neutron star binaries in which massive helium stars, which had their hydrogen envelope removed during a common-envelope phase, remain compact and avoid mass transfer onto the neutron star companion, possibly avoiding pulsar recycling. We present three-dimensional simulations of the supernova explosion of the massive stripped helium star and follow the mass fallback evolution and the subsequent accretion onto the neutron star companion. We find that fallback leads to significant mass growth in the newly formed neutron star. This can explain the formation of heavy binary neutron star systems such as GW190425, as well as predict the assembly of light black hole-neutron star systems such as GW200115. This formation avenue is consistent with the observed mass-eccentricity correlation of binary neutron stars in the Milky Way. Finally, avoiding mass transfer suggests an unusually long spin-period population of pulsar binaries in our Galaxy.