Boron nitride (BN) is today considered one of the most promising materials for many novel applications including bright single photon emission, deep UV optoelectronics, small-sized solid-state neutron detectors, high-performance two-dimensional materials, etc. Recently, AA' stacking a few layers (2-5 layers) of BN films and ABC stacking multilayer (2.2-12 nm) BN films have been successfully synthesized, respectively. However, the preparation of single crystal metal foil and the subsequent growth of BN thin films are undoubtedly incompatible with the current semiconductor process. In addition, how to achieve an accurate layer-by-layer connection between adjacent multilayer BN domains still needs further research. In this work, we demonstrate an approach to grow large-area multilayer single-crystal BN films by chemical vapor deposition on face-centered cubic Fe-Ni (111) single-crystal alloy thin films with different stoichiometric phases, which are deposited on Sapphire. We show that the BN growth is greatly tunable and improved by increasing the Fe content in single-crystal Fe-Ni (111). The formation of pyramid-shaped multilayer BN domains with aligned orientation enables a continuous connection following a layer-by-layer, "first-meet-first-connect", mosaic stitching mechanism. By means of selected area electron diffraction, microphotoluminescence spectroscopy in the deep UV, and high-resolution transmission electron microscopy, the layer-by-layer connection mechanism is unambiguously evidenced, and the stacking order has been verified to occur as unidirectional AB and ABC stackings, i.e., in the Bernal and rhombohedral BN phase.