Millimeter-wave (mmWave) OFDM radar equipped with rainbow beamforming, enabled by phase-time arrays (PTAs), provides wide-angle coverage and is well-suited for fast real-time target detection and tracking. However, accurate detection of multiple closely spaced targets remains a key challenge for conventional signal processing pipelines, particularly those relying on constant false alarm rate (CFAR) detectors. This paper presents CFARNet, a learning-based processing framework that replaces CFAR with a convolutional neural network (CNN) for peak detection in the angle-Doppler domain. The network predicts target subcarrier indices, which guide angle estimation via a known frequency-angle mapping and enable high-resolution range and velocity estimation using the MUSIC algorithm. Extensive simulations demonstrate that CFARNet significantly outperforms a baseline combining CFAR and MUSIC, especially under low transmit power and dense multi-target conditions. The proposed method offers superior angular resolution, enhanced robustness in low-SNR scenarios, and improved computational efficiency, highlighting the potential of data-driven approaches for high-resolution mmWave radar sensing.