It has been proposed that Earthshine, i.e. light and heat from the molten Earth after the Moon forming impact when the Moon was much closer, heated the near side of the tidally locked Moon and that this caused the differences in crust thickness by allowing different materials accrete and condensate as the Moon cooled. Basically, more of the crusty materials condensated on the colder far side, to put it in the blunt way that I can relate to.
I (not being a scientist) would think that tidal forces played an important part too. Causing different volcanic activities on the near and far sides. The only somewhat similar system is the tidally locked Pluto/Charon system and maybe something could be learned from how their crusts are differentiated from facing and opposing hemispheres.
Mars too has a basic dichotomy in its topography, although not obviously related to any companion present or past.