Graphene, as a semimetal with the largest known thermal conductivity, is an ideal system to study the interplay between electronic and lattice contributions to thermal transport. While the total electrical and thermal conductivity have been extensively investigated, a detailed first-principles study of its electronic thermal conductivity is still missing. Here, we first characterize the electron phonon intrinsic contribution to the electronic thermal resistivity of graphene as a function of doping using electronic and phonon dispersions and electron phonon couplings calculated from first principles at the level of density-functional theory and many-body perturbation theory (GW). Then, we include extrinsic electron impurity scattering using low-temperature experimental estimates. Under these conditions, we find that the in-plane electronic thermal conductivity ice of doped graphene is 300 W/mK at room temperature, independently of doping. This result is much larger than expected and comparable to the total thermal conductivity of typical metals, contributing 40% to the total thermal conductivity of bulk graphene. Notably, in samples whose physical or domain sizes are of the order of few micrometers or smaller, the relative contribution coming from the electronic thermal conductivity is more important than in the bulk limit, because lattice thermal conductivity is much more sensitive to sample or grain size at these scales. Last, when electron-impurity scattering effects are included we find that the electronic thermal conductivity is reduced by 30 to 70%. We also find that the Wiedemann Franz law is broadly satisfied at low and high temperatures but with the largest deviations of 20-50% around room temperature.