  This work presents a new constructive uniqueness proof for Calder\'on's
inverse problem of electrical impedance tomography, subject to local Cauchy
data, for a large class of piecewise constant conductivities that we call
"piecewise constant layered conductivities" (PCLC). The resulting
reconstruction method only relies on the physically intuitive monotonicity
principles of the local Neumann-to-Dirichlet map, and therefore the method
lends itself well to efficient numerical implementation and generalization to
electrode models. Several direct reconstruction methods exist for the related
problem of inclusion detection, however they share the property that "holes in
inclusions" or "inclusions-within-inclusions" cannot be determined. One such
method is the monotonicity method of Harrach, Seo, and Ullrich, and in fact the
method presented here is a modified variant of the monotonicity method which
overcomes this problem. More precisely, the presented method abuses that a PCLC
type conductivity can be decomposed into nested layers of positive and/or
negative perturbations that, layer-by-layer, can be determined via the
monotonicity method. The conductivity values on each layer are found via basic
one-dimensional optimization problems constrained by monotonicity relations.
