Motivated by the recent successful formation of boron and nitrogen-doped graphene, using density-functional-theory calculations, we investigate the atomic and electronic structure of bilayers of a graphene-like borocarbonitride ($\mathrm{g}\text{\ensuremath{-}}{\mathrm{BC}}_{6}\mathrm{N}@2\mathrm{L}$). Although pristine graphene is a semimetal, the single layer of $\mathrm{g}\text{\ensuremath{-}}{\mathrm{BC}}_{6}\mathrm{N}$ is a semiconductor with a direct band gap of 1.3 eV, as is the bilayer $\mathrm{g}\text{\ensuremath{-}}{\mathrm{BC}}_{6}\mathrm{N}@2\mathrm{L}$, which has a slightly smaller band gap of 1.2 eV. Systematic calculations for the bilayer structure, to investigate the effect of an electric field (E field), reveal that with increase of E-field, from 0.1 to 0.6 (V/\AA{}), the band gap decreases linearly for both parallel and antiparallel E-field directions. For larger E fields ($g0.6$ V/\AA{}), we find the formation of dual narrow band gaps of 50 meV (for parallel) and 0.4 eV (for antiparallel). The shape of energy bands is preserved and the value of band gaps are constant in both E-field directions. Our results indicate that monolayer and bilayer graphene-like borocarbonitride ($\mathrm{g}\text{\ensuremath{-}}{\mathrm{BC}}_{6}\mathrm{N}$) is an interesting system which may find applications in future low-dissipation, high-speed nanoelectronic devices.