Hot, Dust-Obscured Galaxies (Hot DOGs), selected from the WISE all sky infrared survey, host some of the most powerful Active Galactic Nuclei (AGN) known, and might represent an important stage in the evolution of galaxies. Most known Hot DOGs are at $z> 1.5$, due in part to a strong bias against identifying them at lower redshift related to the selection criteria. We present a new selection method that identifies 153 Hot DOG candidates at $zsim 1$, where they are significantly brighter and easier to study. We validate this approach by measuring a redshift $z=1.009$, and an SED similar to higher redshift Hot DOGs for one of these objects, WISE J1036+0449 ($L_{rm,Bol}simeq 8times 10^{46}rm,erg,s^{-1}$), using data from Keck/LRIS and NIRSPEC, SDSS, and CSO. We find evidence of a broadened component in MgII, which, if due to the gravitational potential of the supermassive black hole, would imply a black hole mass of $M_{rm,BH}simeq 2 times 10^8 M_{odot}$, and an Eddington ratio of $lambda_{rm,Edd}simeq 2.7$. WISE J1036+0449 is the first Hot DOG detected by NuSTAR, and the observations show that the source is heavily obscured, with a column density of $N_{rm,H}simeq(2-15)times10^{23}rm,cm^{-2}$. The source has an intrinsic 2-10 keV luminosity of $sim 6times 10^{44}rm,erg,s^{-1}$, a value significantly lower than that expected from the mid-infrared/X-ray correlation. We also find that the other Hot DOGs observed by X-ray facilities show a similar deficiency of X-ray flux. We discuss the origin of the X-ray weakness and the absorption properties of Hot DOGs. Hot DOGs at $zlesssim1$ could be excellent laboratories to probe the characteristics of the accretion flow and of the X-ray emitting plasma at extreme values of the Eddington ratio.
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