  We present spatially resolved measurements of cool gas flowing into and out
of the Milky Way (MW), using archival ultraviolet spectra of background quasars
from the Hubble Space Telescope/Cosmic Origins Spectrograph. We co-add spectra
of different background sources at close projected angular separation on the
sky. This novel stacking technique dramatically increases the signal-to-noise
ratio of the spectra, allowing detection of low column density gas (down to
$EW$ > 2 mA). We identify absorption as inflowing or outflowing, by using
blue/redshifted high velocity cloud (HVC) absorption components in the
Galactocentric rest frame, respectively. The mass surface densities of
inflowing and outflowing gas both vary by more than an order of magnitude
across the sky, with mean values of $\langle \Sigma_{in}\rangle \gtrsim
10^{4.6\pm0.1}$ $M_{\odot}\,\mathrm{kpc}^{-2}$ for inflowing gas and $\langle
\Sigma_{out}\rangle \gtrsim 10^{3.5\pm 0.1}$ $M_{\odot}\,\mathrm{kpc}^{-2}$ for
outflowing gas, respectively. The mass flow rate surface densities (mass flow
rates per unit area) also show large variation across the sky with $\langle
\dot{\Sigma}(d)_{in}\rangle \gtrsim (10^{-3.6\pm0.1})(d/12 \mathrm{kpc})^{-1}
M_{\odot}\,\mathrm{kpc}^{-2} \mathrm{yr}^{-1}$ for inflowing and $\langle
\dot{\Sigma}(d)_{out}\rangle \gtrsim (10^{-4.8\pm0.1})(d/12\,\mathrm{kpc})^{-1}
M_{\odot}\,\mathrm{kpc}^{-2}\,\mathrm{yr}^{-1}$ for outflowing gas,
respectively. The regions with highest surface mass density of inflowing gas
are clustered at smaller angular scales ($\theta < 40^\circ$). This indicates
that most of the mass in inflowing gas is confined to small, well-defined
structures, whereas the distribution of outflowing gas is spread more uniformly
throughout the sky. Our study confirms that the MW is predominantly accreting
gas, but is also losing a non-negligible mass of gas via outflow.
