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 Öttinger'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.
Keywords: GENERIC formalism; extended irreversible thermodynamics; heat conduction; kinetic theory; relativistic hydrodynamics.