In this paper, we present a side-channel attack on the hardware AES accelerator of a Bluetooth chip used in millions of devices worldwide, ranging from wearables and smart home products to industrial IoT. The attack leverages information about AES computations unintentionally transmitted by the chip together with RF signals to recover the encryption key. Unlike traditional side-channel attacks that rely on power or near-field electromagnetic emissions as sources of information, RF-based attacks leave no evidence of tampering, as they do not require package removal, chip decapsulation, or additional soldered components. However, side-channel emissions extracted from RF signals are considerably weaker and noisier, necessitating more traces for key recovery. The presented profiled machine learning-assisted attack can recover the full encryption key from 45,000 traces captured at a one-meter distance from the target device, with each trace being an average of 10,000 samples per encryption. This is a fourfold improvement over the correlation analysis-based attack on the same AES accelerator.