  We treat an inverse electrical conductivity problem which deals with the
reconstruction of nonlinear electrical conductivity starting from boundary
measurements in steady currents operations. In this framework, a key role is
played by the Monotonicity Principle, which establishes a monotonic relation
connecting the unknown material property to the (measured) Dirichlet-to-Neumann
operator (DtN). Monotonicity Principles are the foundation for a class of
non-iterative and real-time imaging methods and algorithms. In this article, we
prove that the Monotonicity Principle for the Dirichlet Energy in nonlinear
problems holds under mild assumptions. Then, we show that apart from linear and
$p$-Laplacian cases, it is impossible to transfer this Monotonicity result from
the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce
a new boundary operator, identified as an Average DtN operator.
