In this letter, we present a single-pixel compressive direction of arrival (DoA) estimation technique leveraging a graph attention network (GAT)-based deep-learning framework. The physical layer compression is achieved using a coded-aperture technique, probing the spectrum of far-field sources that are incident on the aperture using a set of spatio-temporally incoherent modes. This information is then encoded and compressed into the channel of the coded-aperture. The coded-aperture is based on a metasurface antenna design and it works as a receiver, exhibiting a single-channel and replacing the conventional multi-channel raster scan-based solutions for DoA estimation. The GAT network enables the compressive DoA estimation framework to learn the DoA information directly from the measurements acquired using the coded-aperture. This step eliminates the need for an additional reconstruction step and significantly simplifies the processing layer to achieve DoA estimation. We show that the presented GAT integrated single-pixel radar framework can retrieve high fidelity DoA information even under relatively low signal-to-noise ratio (SNR) levels.