We develop a mesoscopic model to study the plastic behavior of an amorphous material under cyclic loading. The model is depinning-like and driven by a disordered thresholds dynamics that is coupled by long-range elastic interactions. We propose a simple protocol of “glass preparation” that allows us to mimic thermalization at high temperatures as well as aging at vanishing temperature. Various levels of glass stabilities (from brittle to ductile) can be achieved by tuning the aging duration. The aged glasses are then immersed into a quenched disorder landscape and serve as initial configurations for various protocols of mechanical loading by shearing. The dependence of the plastic behavior upon monotonous loading is recovered. The behavior under cyclic loading is studied for different ages and system sizes. The size and age dependence of the irreversibility transition is discussed. A thorough characterization of the disorder-landscape is achieved through the analysis of the transition graphs, which describe the plastic deformation pathways under athermal quasi-static shear. In particular, the analysis of the stability ranges of the strongly connected components of the transition graphs reveals the emergence of a phase-separation like process associated with the aging of the glass. Increasing the age and, hence, the stability of the initial glass results in a gradual break-up of the landscape of dynamically accessible stable states into three distinct regions: one region centered around the initially prepared glass phase and two additional regions characterized by well-separated ranges of positive and negative plastic strains, each of which is accessible only from the initial glass phase by passing through the stress peak in the forward and backward, respectively, shearing directions.