Sudden Stratospheric Warmings (SSWs) are key sources of subseasonal predictability and major drivers of extreme weather in winter. Accurate and efficient probabilistic forecasting of these events remains a persistent challenge for Numerical Weather Prediction (NWP) systems due to computational bottlenecks and limitations in physical representation. While data-driven forecasting is rapidly evolving, its application to the complex, three-dimensional dynamics of SSWs remains underexplored. Here, we bridge this gap by developing a Flow Matching-based generative AI model (FM-Cast) for efficient and skillful probabilistic forecasting of the spatiotemporal evolution of stratospheric circulation in winter. Evaluated across 18 major SSW events (1998-2024), FM-Cast successfully forecasts the onset, intensity, and 3D morphology of the polar vortex up to 15 days in advance for most cases. Notably, it achieves long-range probabilistic forecast skill comparable to or exceeding leading operational NWP systems (ECMWF and CMA) while generating a 30-day forecast with 50-member ensemble, in just two minutes on a consumer GPU. Furthermore, using idealized "perfect troposphere" experiments, we uncover distinct predictability regimes: events driven by continuous wave forcing versus those governed by an initial trigger and subsequent stratospheric dynamical memory. This work establishes a computationally efficient paradigm for probabilistic stratospheric forecasting that simultaneously deepens our physical understanding of atmosphere-climate dynamics.