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A procedure for the numerical approximation of high-dimensional Hamilton--Jacobi--Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the successive Galerkin approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen.