We consider the computation of Taylor approximations to the optimal feedback law and value function for infinite-horizon finite-dimensional nonlinear optimal control problem --- also known as Al'brekht's method. The computational time scalability has been partially mitigated by recent works leveraging tensor-structure to develop efficient numerical methods. However, RAM requirements still have remained a major hindrance to scaling beyond state dimensions of a few hundred or low thousands. We propose an acceleration to compute the expensive higher-order terms in a reduced space, mitigating the curse of dimensionality. The provided open-source Matlab implementation is designed for general purpose use, accomodating arbitrary polynomial nonlinearities in the control-affine dynamics, polynomial state nonlinearities in the cost function, and the ability to handle systems in generalized form with a non-identity mass matrix, such as finite element models. Combined with the exploitation of sparsity and modern low-rank Riccati solvers, this permits computing approximations to polynomial feedback laws based on Al'brekht's method in hundreds of thousands of states, as demonstrated on various examples in comparison with other state-of-the-art alternative methods.