We propose a snapshot spectral imaging method for the visible spectral range using a single monochromatic camera equipped with a two-dimensional (2D) binary-encoded phase diffuser placed at the pupil of the imaging lens and by resorting to deep learning (DL) algorithms for signal reconstruction. While spectral imaging was shown to be feasible using two cameras equipped with a single, one-dimensional (1D) binary diffuser and compressed sensing (CS) algorithms [Appl. Opt.59, 7853 (2020).APOPAI0003-693510.1364/AO.395541], the suggested diffuser design expands the optical response and creates optical spatial and spectral encoding along both dimensions of the image sensor. To recover the spatial and spectral information from the dispersed and diffused (DD) monochromatic snapshot, we developed novel DL algorithms, dubbed DD-Nets, which are tailored to the unique response of the optical system, which includes either a 1D or a 2D diffuser. High-quality reconstructions of the spectral cube in simulation and lab experiments are presented for system configurations consisting of a single monochromatic camera with either a 1D or a 2D diffuser. We demonstrate that the suggested system configuration with the 2D diffuser outperforms system configurations with a 1D diffuser that utilize either DL-based or CS-based algorithms for the reconstruction of the spectral cube.