  We present an atmospheric retrieval analysis of HST/WFC3/G141 spectroscopic
phase curve observations of two brown dwarfs, WD-0137B and EPIC-2122B, in
ultra-short period orbits around white dwarf hosts. These systems are analogous
to hot and ultra-hot Jupiter systems, enabling a unique and high-precision
comparison to exoplanet systems. We use the PETRA retrieval suite to test
various analysis setups, including joint-phase retrievals, multiple temperature
structures, and non-uniform abundances. We find that WD-0137B has a dayside
that closely resembles that of other ultra-hot Jupiters with inverted
temperature structures and H$^-$ opacity, but quickly transitions to a mostly
non-inverted temperature structure on the nightside. Meanwhile, EPIC-2122B's
atmosphere remains inverted at all constrained longitudes, with dominant H$^-$
opacity. Retrievals with multiple temperature profiles and non-uniform vertical
abundances were generally not statistically justified for this dataset, but
retrievals with dayside-dilution factors were found to be justified. Retrieving
all phases simultaneously with a linear combination of a dayside and nightside
atmosphere was found to be an adequate representation of the entire phase-curve
once a longitudinal temperature gradient free parameter was included in the
retrieval. Comparing to global circulation models, we attribute behavior in the
1D retrievals to the inclined viewing geometry of the systems, which results in
always-visible irradiated and inverted portions of the atmosphere
"contaminating" spectra measured from the nightside hemisphere. This study
sheds light on the similarities between these irradiated brown dwarf systems
and hot and ultra-hot Jupiters, but also their unique differences, including
the influence of the inclined viewing geometry.
