  We propose a general procedure for evaluating, directly from microphysics,
the constitutive relations of heat-conducting fluids in regimes of large fluxes
of heat. Our choice of hydrodynamic formalism is Carter's two-fluid theory,
which happens to coincide with \"{O}ttinger's GENERIC theory for relativistic
heat conduction. This is a natural framework, as it should correctly describe
the relativistic ``inertia of heat'' as well as the subtle interplay between
reversible and irreversible couplings. We provide two concrete applications of
our procedure, where the constitutive relations are evaluated respectively from
maximum entropy hydrodynamics and Chapman-Enskog theory.
