  The spectra of high-redshift ($z\gtrsim 6$) quasars contain valuable
information on the progression of the Epoch of Reionization (EoR). At redshifts
$z<6$, the observed Lyman-series forest shows that the intergalactic medium
(IGM) is nearly ionized, while at $z>7$ the observed quasar damping wings
indicate high neutral gas fractions. However, there remains a gap in neutral
gas fraction constraints at $6\lesssim z \lesssim 7$ where the Lyman series
forest becomes saturated but damping wings have yet to fully emerge. In this
work, we use a sample of 18 quasar spectra at redshifts $6.0<z<7.1$ to close
this gap. We apply neural networks to reconstruct the quasars' continuum
emission around the partially absorbed Lyman $\alpha$ line to normalize their
spectra, and stack these continuum-normalized spectra in three redshift bins.
To increase the robustness of our results, we compare the stacks to a grid of
models from two hydrodynamical simulations, ATON and CROC, and we measure the
volume-averaged neutral gas fraction, $\bar{x}_{\rm HI}$, while jointly fitting
for the mean quasar lifetime, $t_{\rm Q}$, for each stacked spectrum. We
chronicle the evolution of neutral gas fraction using the ATON (CROC) models as
follows: $\bar{x}_{\rm HI} = 0.21_{-0.07}^{+0.17}$ ($\bar{x}_{\rm HI} =
0.10_{<10^{-4}}^{+0.73}$) at $\langle z \rangle =6.10$, $\bar{x}_{\rm HI} =
0.21_{-0.07}^{+0.33}$ ($\bar{x}_{\rm HI} =0.57_{-0.47}^{+0.26}$) at $\langle z
\rangle =6.46$, and $\bar{x}_{\rm HI} = 0.37_{-0.17}^{+0.17}$ ($\bar{x}_{\rm
HI} =0.57_{-0.21}^{+0.26}$) at $\langle z \rangle =6.87$. At the same time we
constrain the average quasar lifetime to be $t_{\rm Q} \lesssim 7\ {\rm Myr}$
across all redshift bins, in good agreement with previous studies.
