Fibre-reinforced laminated composites are constructed layer-by-layer, enabling ease of directional stiffness tailoring. Their vast design space is typically explored using two-steps. First, the optimum stiffness for given loads is conceptualised using continuous optimisation of lamination parameters (LPs). Then, discrete optimisation determines a fibre stacking sequence (SS) that closely matches these LPs. While fibre angles are conventionally limited to 45°multiples, finer increments (e.g., ±15°) can enable lighter structures. However, existing SS design methods do not scale well with this increased problem dimensionality. To overcome this challenge, we propose LP2SS, a novel methodology utilising fast Fourier transforms (FFT) and a branch-and-bound optimiser. By treating LPs as a signal, FFTs identify the number of fibre layers oriented at different angles, akin to estimating the magnitude of different frequencies within a signal. This fibre angle distribution guides the branch-and-bound optimiser, enabling efficient SS design with accurate LP matching, while satisfying empirical design rules. The ingenious use of FFTs is key to LP2SS's performance, achieving solutions within tenths of a second, compared to minutes required by state-of-the-art methods. Validated on established benchmarks and a newly proposed comprehensive test set, LP2SS marks a significant advancement in the optimal design of large-scale laminated composite structures.