When exposed to strong shearing, the particles in a crystal will rearrange and ultimately, the crystal will break by forming large nonaffine defects. Even for the initial stage of this process, only little effort has been devoted to the understanding of the breaking process on the scale of the individual particle size for thermalized mixed crystals. Here, we explore the shear-induced breaking for an equimolar two-dimensional binary model crystal with a high interaction asymmetry between the two different species such that the initial crystal has an intersecting square sublattice of the two constituents. Using Brownian dynamics computer simulations, we show that the combination of shear and thermal fluctuations leads to a characteristic hierarchical breaking scenario where initially, the more strongly coupled particles are thermally distorted, paving the way for the weakly coupled particles to escape from their cage. This in turn leads to mobile defects which may finally merge, proliferating a cascade of defects, which triggers the final breakage of the crystal. This scenario is in marked contrast to the breakage of one-component crystals close to melting. Moreover, we explore the orientational dependence of the initial shear direction relative to the crystal orientation and compare this to the usual melting scenario without shear. Our results are verifiable in real-space experiments of superparamagnetic colloidal mixtures at a pending air-water interface in an external magnetic field where the shear can be induced by an external laser field.