High-gain beamforming (BF) is essential for low Earth orbit (LEO) satellite communications to overcome severe path loss, but this requires acquiring precise satellite positions. Conventional satellite acquisition typically relies on time-domain beam sweeping, which incurs substantial overhead and latency. In this correspondence, we propose an efficient one-shot satellite acquisition framework that capitalizes on two phenomena traditionally regarded as impairments: i) Doppler effects and ii) beam-squint effects. Specifically, we derive a closed-form \emph{rainbow beamformer} that leverages beam-squint effects to align frequency-dependent beam directions with satellite positions inferred from their Doppler shifts. This approach enables reception from multiple satellites at once without requiring beam sweeping. To extract satellite position information, we develop three Doppler-aware angle estimation algorithms based on received signals. Simulation results demonstrate that the proposed method significantly outperforms conventional beam sweeping approaches in both acquisition accuracy and required time slots. These gains stem from the ability of the proposed rainbow BF to exploit the \emph{angle-dependent nature of Doppler shifts}, enabling full angular-domain coverage with a single pilot transmission and reception.