Herein, we perform DFT calculations to study the stability and the electronic and optical properties of pristine and hybrid bilayers of hexagonal boron phosphide (h-BP) and hexagonal boron arsenide (h-BAs). The electronic and optical properties of all bilayers can be modulated by the introduction of an external perpendicular electric field. Consequently, band gaps can be tuned from 0.8 eV down to zero as the field increases up to a critical value. Above this value, the gap reopens as the valence and conduction bands exhibit an anticrossing with a "Mexicanhat" shape, in a similar fashion to biased graphene bilayers. For the optical properties, we report an intense peak in the absorption spectra around 2.5-2.6 eV, with a slight blueshift as a function of electric field. Above the critical field, a new peak is observed in the infrared region which exhibits a strong field dependence. Additionally, this peak is related to the optical transitions around the "Mexican-hat" region of the band structure and the corresponding electron-hole pair shows a layer separation, which may lead to larger recombination times, a critical factor in photovoltaics. We believe that, with such tunable properties, these bilayers can find interesting applications in future devices in nanoelectronics and optoelectronics.