In joint works with Kevin Yang, we consider a stochastic Laplacian growth model, that can be viewed as a continuum version of origin-excited random walks. Here, we grow the (d+1)-dimensional manifold M(t) according to a reflecting Brownian motion (RBM) on M(t), stopped at level sets of its boundary local time. An averaging principle for the RBM characterizes the scaling limit for the leading order behavior of the interface (namely, the boundary of M(t)). This limit is given by a locally well-posed, geometric flow-type PDE, whose blow-up times correspond to changes in the diffeomorphism class of the growing set. Smoothing the interface as we inflate M(t), yields an SPDE for the large-scale fluctuations of an associated height function. This SPDE is a regularized KPZ-type equation, modulated by a Dirichlet-to-Neumann operator. For d=1 we can further remove the regularization, so the fluctuations of M(t) now have a double-scaling limit given by a singular KPZ-type equation.