We explore the potential of a simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) to enhance the performance of wireless surveillance systems. The STAR-RIS is deployed between a full-duplex (FD) multi-antenna legitimate eavesdropper (E) and a suspicious communication pair. It reflects the suspicious signal towards the suspicious receiver (SR), while simultaneously transmitting the same signal to E for interception purposes. Additionally, it enables the forwarding of a jamming signal from E to SR, which is located on the back side of the STAR-RIS. To enhance the eavesdropping non-outage probability, we formulate a non-convex joint optimization problem to design the beamforming vectors at E and reflection/transmission phase shift matrices at the STAR-RIS. We adopt the block coordinate descent (BCD) algorithm and propose an approach, mainly based on semi-definite relaxation (SDR) and successive convex approximation (SCA), for solving the resulting decoupled sub-problems. Finally, we compare the performance of the proposed design against low-complexity zero-forcing (ZF)-based beamforming designs.