  Bulk quantum fields are often said to contribute to the generalized entropy
$\frac{A}{4G_N} +S_{\mathrm{bulk}}$ only at $O(1)$. Nonetheless, in the context
of evaporating black holes, $O(1/G_N)$ gradients in $S_{\mathrm{bulk}}$ can
arise due to large boosts, introducing a quantum extremal surface far from any
classical extremal surface. We examine the effect of such bulk quantum effects
on quantum extremal surfaces (QESs) and the resulting entanglement wedge in a
simple two-boundary $2d$ bulk system defined by Jackiw-Teitelboim gravity
coupled to a 1+1 CFT. Turning on a coupling between one boundary and a further
external auxiliary system which functions as a heat sink allows a two-sided
otherwise-eternal black hole to evaporate on one side. We find the generalized
entropy of the QES to behave as expected from general considerations of
unitarity, and in particular that ingoing information disappears from the
entanglement wedge after a scambling time $\frac{\beta}{2\pi} \ln \Delta S +
O(1)$ in accord with expectations for holographic implementations of the
Hayden-Preskill protocol. We also find an interesting QES phase transition at
what one might call the Page time for our process.
