  We present novel constraints on the underlying galaxy formation physics
(e.g., mass loading factor, star formation history, metal retention) at
$z\gtrsim7$ for the low-mass ($M_*\sim10^5$ M$_\odot$) Local Group ultra-faint
dwarf galaxy (UFD) Eridanus {\sc II} (Eri II). Using a hierarchical Bayesian
framework, we apply a one-zone chemical evolution model to Eri II's CaHK-based
photometric metallicity distribution function (MDF; [Fe/H]) and find that the
evolution of Eri II is well-characterized by a short, exponentially declining
star-formation history ($\tau_\text{SFH}=0.39\pm_{0.13}^{0.18}$ Gyr), a low
star-formation efficiency ($\tau_\text{SFE}=27.56\pm_{12.92}^{25.14}$ Gyr), and
a large mass-loading factor ($\eta=194.53\pm_{42.67}^{33.37}$). Our results are
consistent with Eri II forming the majority of its stars before the end of
reionization. The large mass-loading factor implies strong outflows in the
early history of Eri II and is in good agreement with theoretical predictions
for the mass-scaling of galactic winds. It also results in the ejection of
$>$90\% of the metals produced in Eri II. We make predictions for the
distribution of [Mg/Fe]-[Fe/H] in Eri II as well as the prevalence of ultra
metal-poor stars, both of which can be tested by future chemical abundance
measurements. Spectroscopic follow-up of the highest metallicity stars in Eri
II ($\text{[Fe/H]} > -2$) will greatly improve model constraints. Our new
framework can readily be applied to all UFDs throughout the Local Group,
providing new insights into the underlying physics governing the evolution of
the faintest galaxies in the reionization era.
