Hot dust-obscured galaxies (hot DOGs), selected from Wide-Field Infrared Survey Explorer's all-sky infrared survey, host some of the most powerful active galactic nuclei known and may represent an important stage in the evolution of galaxies. Most known hot DOGs are located 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 similar to 1, where they are significantly brighter and easier to study. We validate this approach by measuring a redshift z = 1.009 and finding a spectral energy distribution similar to that of higher-redshift hot DOGs for one of these objects, WISE J1036+ 0449 (L-Bol similar or equal to 8 x 10(46) erg s(-1)). We find evidence of a broadened component in Mg II, which would imply a black hole mass of M-BH similar or equal to 2 x 10(8) M-circle dot and an Eddington ratio of lambda(Edd) similar or equal to 2.7. WISE J1036+ 0449 is the first hot DOG detected by the Nuclear Spectroscopic Telescope Array, and observations show that the source is heavily obscured, with a column density of N-II similar or equal to (2-15) x 10(23)cm(-2). The source has an intrinsic 2-10 keV luminosity of similar to 6 x 10(44)erg s(-1), a value significantly lower than that expected from the mid-infrared/X-ray correlation. We also find that 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 <= 1 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.