The BN sheet is a nonmagnetic wide-band-gap semiconductor. Using density functional theory, we show that these properties can be fundamentally altered by embedding graphene flakes. Not only do graphene flakes preserve the two-dimensional (2D) planar structure of the BN sheet, but by controlling their shape and size, unexpected electronic and magnetic properties also emerge. The electronic band structure can be tuned from a direct gap to an indirect gap, the energy gap can be further modulated by changing the bonding patterns, and both hole injecting or electron injecting can be achieved by tailoring the triangular embedding pattern. Furthermore, the Lieb theorem still holds, and the embedded triangular graphene flakes become ferromagnetic with full spin polarizations of the introduced electrons or holes, opening the door to their use as spin filters. The study sheds new light on hybrid single-atomic-layer engineering for unprecedented applications of 2D nanomaterials.