  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 $z\sim
1$, 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
($L_{\rm\,Bol}\simeq 8\times 10^{46}\rm\,erg\,s^{-1}$), 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 $\sim 6\times 10^{44}\rm\,erg\,s^{-1}$, 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 $z\lesssim1$ 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.
