  Early star-forming galaxies produced copious ionizing photons. A fraction of
these photons escaped gas within galaxies to reionize the entire Universe. This
escape fraction is crucial for determining how the Universe became reionized,
but the neutral intergalactic medium precludes direct measurement of the escape
fraction at high-redshifts. Indirect estimates of the escape fraction must
describe how the Universe was reionized. Here, we present new Keck Cosmic Web
Imager spatially-resolved spectroscopy of the resonant Mg II 2800 doublet from
a redshift 0.36 galaxy, J1503+3644, with a previously observed escape fraction
of 6%. The Mg II emission has a similar spatial extent as the stellar
continuum, and each of the Mg II doublet lines are well-fit by single
Gaussians. The Mg II is optically thin. The intrinsic flux ratio of the red and
blue Mg II emission line doublet, $R = F_{2796}/F_{2803}$, is set by atomic
physics to be two, but Mg$^+$ gas along the line of sight decreases $R$
proportional to the Mg II optical depth. Combined with the metallicity, $R$
estimates the neutral gas column density. The observed $R$ ranges across the
galaxy from 0.8-2.7, implying a factor of 2 spatial variation of the relative
escape fraction. All of the ionizing photons that escape J1503+3644 pass
through regions of high $R$. We combine the Mg II emission and dust attenuation
to accurately estimate the absolute escape fractions for ten local Lyman
Continuum emitting galaxies and suggest that Mg II can predict escape fraction
within the Epoch of Reionization.
