Our closest Sun-like star, Alpha Cen A, is an exceptional target for direct imaging. It offers a nearly 3-fold improvement in the angular scale of its habitable zone (HZ) and a 7.5-fold boost in planet brightness compared to the next nearest G-type star Tau Ceti. JWST's highly sensitive mid-infrared imaging capabilities complement existing radial velocity constraints and enable the direct detection of >5 Earth radii gas giant planets within Alpha Cen A's HZ, including potentially confirming the VLT/NEAR C1 candidate. A Cycle 1 program using the MIRI F1550C coronagraph was executed in August 2024 and probed the 1-3 au region around Alpha Cen A where planets can survive the destabilizing influence of Alpha Cen B. In this talk, I will provide a comprehensive summary of this program's results and place them in the context of our understanding of planet formation in binary systems as well as opportunities with next-generation instruments (e.g., EELT). I will discuss the extensive preparations required for these novel observations, detail our approach to suppressing Alpha Cen A and B's starlight, and address the question: does our nearest Sun-like star host a giant planet?