In this work, we address the problem of eavesdropping on digital video displays by analyzing the electromagnetic waves that unintentionally emanate from the cables and connectors, particularly HDMI. This problem is known as TEMPEST. Compared to the analog case (VGA), the digital case is harder due to a 10-bit encoding that results in a much larger bandwidth and non-linear mapping between the observed signal and the pixel's intensity. As a result, eavesdropping systems designed for the analog case obtain unclear and difficult-to-read images when applied to digital video. The proposed solution is to recast the problem as an inverse problem and train a deep learning module to map the observed electromagnetic signal back to the displayed image. However, this approach still requires a detailed mathematical analysis of the signal, firstly to determine the frequency at which to tune but also to produce training samples without actually needing a real TEMPEST setup. This saves time and avoids the need to obtain these samples, especially if several configurations are being considered. Our focus is on improving the average Character Error Rate in text, and our system improves this rate by over 60 percentage points compared to previous available implementations. The proposed system is based on widely available Software Defined Radio and is fully open-source, seamlessly integrated into the popular GNU Radio framework. We also share the dataset we generated for training, which comprises both simulated and over 1000 real captures. Finally, we discuss some countermeasures to minimize the potential risk of being eavesdropped by systems designed based on similar principles.