  We consider 1D integrable systems supporting ballistic propagation of
excitations, perturbed by a localised defect that breaks most conservation laws
and induces chaotic dynamics. Focusing on classical systems, we study an
out-of-equilibrium protocol engineered activating the defect in an initially
homogeneous and far from the equilibrium state. We find that large enough
defects induce full thermalisation at their center, but nonetheless the
outgoing flow of carriers emerging from the defect is non-thermal due to a
generalization of the celebrated Boundary Thermal Resistance effect, occurring
at the edges of the chaotic region. Our results are obtained combining
ab-initio numerical simulations for relatively small-sized defects, with the
solution of the Boltzmann equation, which becomes exact in the scaling limit of
large, but weak defects.
