We consider 2D lattices which are disordered by an external field, in this case by a dense, random distribution of attractive pinning centers of an underlying substrate. The preferred configuration of the 2D system with screened-Coulomb two-body interactions and a fixed number of particles is a triangular lattice configuration. The pinning centers strongly affect the triangular lattice configuration. By keeping the location and the density of the pinning centers fixed, we study the effects of pinning strength on the structural properties of the lattice. We find that the strongly disordered lattice prefers to form irregular domains of size s which are composed entirely of topological defects. The distribution of sizes of these domains, D(s) suggest that s-τ for sufficiently large s. Our results are consistent with τ ≥ 2 in the solid phase and τ < 2 in the liquid phase. We present a general argument in support of our findings and suggest that the physics which dictates the structure of 2D lattices on substrates with randomly distributed pinning centers is the same as that for a broad range of problems in which an external field strongly influences the behavior of an interacting physical system.