  We consider 1D integrable systems supporting ballistic propagation of
quasiparticles, perturbed by a localised defect that breaks most conservation
laws and induces chaotic dynamics. 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.
