LiDAR sensors are critical for autonomous driving perception, yet remain vulnerable to spoofing attacks. Jamming attacks inject high-frequency laser pulses that completely blind LiDAR sensors by overwhelming authentic returns with malicious signals. We discover that while point clouds become randomized, the underlying full-waveform data retains distinguishable signatures between attack and legitimate signals. In this work, we propose PULSAR-Net, capable of reconstructing authentic point clouds under jamming attacks by leveraging previously underutilized intermediate full-waveform representations and simultaneous laser sensing in modern LiDAR systems. PULSAR-Net adopts a novel U-Net architecture with axial spatial attention mechanisms specifically designed to identify attack-induced signals from authentic object returns in the full-waveform representation. To address the lack of full-waveform representations in existing LiDAR datasets under jamming attacks, we introduce a physics-aware dataset generation pipeline that synthesizes realistic full-waveform representations under jamming attacks. Despite being trained exclusively on synthetic data, PULSAR-Net achieves reconstruction rates of 92% and 73% for vehicles obscured by jamming attacks in real-world static and driving scenarios, respectively.