Optimization problems that are constrained by partial differential equations (PDEs) often pose significant computational challenges to black-box optimization algorithms. This is particular the case for non-linear problems that typically rely on multi-query solution of large-scale linearized subsystems. In this regard, this paper proposes a customized solution strategy for a class of semi-linear PDE-constrained optimization problems in annular domains. At its core, the strategy relies on a collection of new Poisson-like preconditioners that are based on boundary-adapted spectral Galerkin methods. The preconditioners are matrix-free and scale linearly with the problem size. To establish proof-of-concept, a case study solves a benchmark control problem for various model parameters. The results show that the preconditioners lead to fast solution strategies that outperform conventional direct approaches.