  We consider the directed mean curvature flow on the plane in a weak Gaussian
random environment. We prove that, when started from a sufficiently flat
initial condition, a rescaled and recentred solution converges to the Cole-Hopf
solution of the KPZ equation. This result follows from the analysis of a more
general system of nonlinear SPDEs driven by inhomogeneous noises, using the
theory of regularity structures. However, due to inhomogeneity of the noise,
the "black box" result developed in the series of works [Hai14, BHZ19, CH16,
BCCH21] cannot be applied directly and requires significant extension to
infinite-dimensional regularity structures.
  Analysis of this general system of SPDEs gives two more interesting results.
First, we prove that the solution of the quenched KPZ equation with a very
strong force also converges to the Cole-Hopf solution of the KPZ equation.
Second, we show that a properly rescaled and renormalised quenched
Edwards-Wilkinson model in any dimension converges to the stochastic heat
equation.
