This letter studies a wireless-powered amplify-and-forward relaying system, where an energy-constrained relay node assists the information transmission from the source to the destination using the energy harvested from the source. We propose a novel two-phase protocol for efficient energy transfer and information relaying, in which the relay operates in full-duplex mode with simultaneous energy harvesting and information transmission. Compared with the existing protocols, the proposed design possesses two main advantages: i) it ensures uninterrupted information transmission since no time switching or power splitting is needed at the relay for energy harvesting; ii) it enables the so-called self-energy recycling, i.e., part of the energy (loop energy) that is used for information transmission by the relay can be harvested and reused in addition to the dedicated energy sent by the source. Under the multiple-input single-output (MISO) channel setup, the optimal power allocation and beamforming design at the relay are derived. Numerical results show a significant throughput gain achieved by our proposed design over the existing time switching-based relay protocol.