This article presents a methodology for the synthesis of finite-dimensional nonlinear output feedback controllers for nonlinear parabolic partial differential equation (PDE) systems with time-dependent spatial domains. Initially, the nonlinear parabolic PDE system is expressed with respect to an appropriate time-invariant spatial coordinate, and a representative (with respect to different initial conditions and input perturbations) ensemble of solutions of the resulting time-varying PDE system is constructed by computing and solving a high-order discretization of the PDE. Then, the Karhunen-Loaève expansion is directly applied to the ensemble of solutions to derive a small set of empirical eigenfunctions (dominant spatial patterns) that capture almost all the energy of the ensemble of solutions. The empirical eigenfunctions are subsequently used as basis functions within a Galerkin model reduction framework to derive low-order ordinary differential equation (ODE) systems that accurately describe the dominant dynamics of the PDE system. The ODE systems are subsequently used for the synthesis of nonlinear output feedback controllers using geometric control methods. The proposed control method is used to stabilize an unstable steady-state of a diffusion-reaction process with nonlinearities, spatially-varying coefficients and time-dependent spatial domain, and is shown to lead to the construction of accurate low-order models and the synthesis of low-order controllers. The performance of the low-order models and controllers is successfully tested through computer simulations.