  The resonantly scattered Lyman-$\alpha$ line illuminates the extended halos
of neutral hydrogen in the circumgalactic medium of galaxies. We present
integral field Keck Cosmic Web Imager observations of double-peaked, spatially
extended Ly$\alpha$ emission in 12 relatively low-mass ($M_{\star} \sim10^9 \,
M_{\odot}$) $z\sim2$ galaxies characterized by extreme nebular emission lines.
Using individual spaxels and small bins as well as radially binned profiles of
larger regions, we find that for most objects in the sample the Ly$\alpha$
blue-to-red peak ratio increases, the peak separation decreases, and the
fraction of flux emerging at line center increases with radius. We use new
radiative transfer simulations to model each galaxy with a clumpy, multiphase
outflow with radially varying outflow velocity, and self-consistently apply the
same velocity model to the low ionization interstellar absorption lines. These
models reproduce the trends of peak ratio, peak separation and trough depth
with radius, and broadly reconcile outflow velocities inferred from Ly$\alpha$
and absorption lines. The galaxies in our sample are well-described by a model
in which neutral, outflowing clumps are embedded in a hotter, more highly
ionized inter-clump medium (ICM), whose residual neutral content produces
absorption at the systemic redshift. The peak ratio, peak separation and trough
flux fraction are primarily governed by the line-of-sight component of the
outflow velocity, the HI column density, and the residual neutral density in
the ICM respectively. Azimuthal asymmetries in the line profile further suggest
non-radial gas motions at large radii and variations in the HI column density
in the outer halos.
