It is show that one can derive a novel BPS bound for the gauged Non-Linear-Sigma-Model (NLSM) Maxwell theory in (3+1) dimensions which can actually be saturated. Such novel bound is constructed using Hamilton-Jacobi equation from classical mechanics. The configurations saturating the bound represent Hadronic layers possessing both Baryonic charge and magnetic flux. However, unlike what happens in the more common situations, the topological charge which appears naturally in the BPS bound is a non-linear function of the Baryonic charge. This BPS bound can be saturated when the surface area of the layer is quantized. The far-reaching implications of these results are discussed. In particular, we determine the exact relation between the magnetic flux and the Baryonic charge as well as the critical value of the Baryonic chemical potential beyond which these configurations become thermodynamically unstable.