We studied stochastic homogenization of a quasi-linear parabolic partial differential equation (PDE) with nonlinear microscopic Robin conditions on a perforated domain. The focus of our work lies in the underlying geometry that does not allow standard stochastic homogenization techniques to be applied directly. Instead, we introduced a concept of regularized homogenization: We proved homogenization on a regularized but still random geometry and demonstrated afterwards that the form of the homogenized equation was independent from the regularization, though the explicit values of the coefficients depended on the regularization. Then, we passed to the regularization limit to obtain the anticipated limit equation where the coefficients were finally independent from the intermediate regularizations. We provided evidence that the regularized homogenization and the classical stochastic homogenization coincided on geometries that indeed allowed stochastic homogenization. Furthermore, we showed that Boolean models of Poisson point processes were covered by our approach.