In this paper, we present a dynamic programming approach for identifying extremal polyomino chains with respect to degree-based topological indices. This approach provides an explicit recurrence and constructive algorithm that enables both the computation of an extremal polyomino chain in linear time with respect to its number of squares, and the enumeration of all maximal configurations in linear time with respect to their amount. As a main application, we resolve a problem posed in 2016 by characterizing the polyomino chains that maximize the Augmented Zagreb Index ($AZI$) for any fixed number of squares. The $AZI$, a degree-based index known for its strong chemical applicability in numerous studies, attains its maximum on two specific families of polyomino chains depending on the parity of their number of squares. We also derive closed-form expressions for the maximum $AZI$ and determine the exact number of extremal configurations. The results presented in this paper are aligned with previous contributions, and establish a constructive methodology for solving extremal problems in chemical graph theory, for which we provide a link to the code in the last section.