  Consider the short-time probability distribution $\mathcal{P}(H,t)$ of the
one-point interface height difference $h(x=0,\tau=t)-h(x=0,\tau=0)=H$ of the
stationary interface $h(x,\tau)$ described by the Kardar-Parisi-Zhang equation.
It was previously shown that the optimal path -- the most probable history of
the interface $h(x,\tau)$ which dominates the upper tail of $\mathcal{P}(H,t)$
-- is described by any of \emph{two} ramp-like structures of $h(x,\tau)$
traveling either to the left, or to the right. These two solutions emerge, at a
critical value of $H$, via a spontaneous breaking of the mirror symmetry
$x\leftrightarrow -x$ of the optimal path, and this symmetry breaking is
responsible for a second-order dynamical phase transition in the system. We
simulate the interface configurations numerically by employing a
large-deviation Monte Carlo sampling algorithm in conjunction with the mapping
between the KPZ interface and the directed polymer in a random potential at
high temperature. This allows us to observe the optimal paths, which determine
each of the two tails of $\mathcal{P}(H,t)$, down to probability densities as
small as $10^{-500}$. At short times we observe mirror-symmetry-broken
traveling optimal paths for the upper tail, and a single mirror-symmetric path
for the lower tail, in good quantitative agreement with analytical predictions.
At long times, even at moderate values of $H$, where the optimal fluctuation
method is \emph{not} supposed to apply, we still observe two well-defined
dominating paths. Each of them violates the mirror symmetry $x\leftrightarrow
-x$ and is a mirror image of the other.
