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Irradiated brown dwarfs (BDs) provide natural laboratories to test our understanding of substellar and irradiated atmospheres. A handful of short-period BDs around white dwarfs (WDs) have been observed, but the uniquely intense UV-dominated irradiation presents a modeling challenge. Here, we present the first fully self-consistent 1D atmosphere models that take into account the UV irradiation's effect on the object's temperature structure. We explore two BD-WD systems, namely WD-0137-349 and EPIC-212235321. WD-0137-349B has an equilibrium temperature that would place it in the transition between hot and ultra-hot Jupiters, while EPIC-212235321B has an equilibrium temperature higher than all ultra-hot Jupiters except KELT-9b. We explore some peculiar aspects of irradiated BD atmospheres and show that existing photometry can be well-fit with our models. Additionally, the detections of atomic emission lines from these BDs can be explained by a strong irradiation-induced temperature inversion, similar to inversions recently explored in ultra-hot Jupiters. Our models of WD-0137-349B can reproduce the observed equivalent width of many but not all of these atomic lines. We use the observed photometry of these objects to retrieve the temperature structure using the PHOENIX ExoplaneT Retrieval Algorithm (PETRA) and demonstrate that the structures are consistent with our models, albeit somewhat cooler at low pressures. We then discuss the similarities and differences between this class of irradiated brown dwarf and the lower-mass ultra-hot Jupiters. Lastly, we describe the behavior of irradiated BDs in color-magnitude space to show the difficulty in classifying irradiated BDs using otherwise well-tested methods for isolated objects.