A novel millimeter-wave design of a Fabry-Perot cavity leaky-wave antenna (LWA) is proposed here as a promising alternative to conventional configurations for improving the directivity at broadside of radiating elements. In order to prove our original idea, the radiation features of i) a substrate-superstrate LWA, ii) a grounded dielectric slab with a single homogenized metasurface above it, and iii) this novel layout, consisting of a substrate-superstrate LWA with interleaved metasurfaces, are evaluated through rigorous analytical methods and validated with reliable numerical techniques. It is seen that the presence of the two metasurfaces significantly improves the directivity of the conventional substrate-superstrate LWA and also leads to a quite higher directivity than the LWA with a single metasurface on top.