We report the mechanism underlying two-step yielding in repulsive colloidal microgel glasses under shear deformation. Strain sweep and start-up flow experiments demonstrate the existence of two-step yielding, which was further investigated by creep-recovery, and Lissajous-Bowditch curves to probe intra-cycle nonlinearities. By increasing the microgel volume fraction, we track the transition from entropic to jammed glass regimes and examine the distinct roles of particle softness and crosslinking heterogeneity in yielding behaviour. Soft core-shell particles exhibit two-step yielding in the jammed glass regime at a frequency of omega = 1 rad s-1. We compare the results for three types of particles: soft core-shell; stiff core-shell; and homogeneously crosslinked. We find that stiff core-shell and homogeneous particles do not exhibit two-step yielding under any experimental conditions. These findings demonstrate that softness combined with a core-shell particle structure is necessary to support two-step yielding. Intra-cycle nonlinearities reveal that strain stiffening develops between the first and second yield points, arising from resistance to macroscopic flow at and beyond the first G `' peak. This resistance to cage breaking originates from the strong interlocking of interpenetrated polymer chains that occurs during microgel deformation and compression in the jammed state. Macroscopic flow begins at the second yield point, where particles escape their cages by breaking the interlocking structure, leading to the G `-G `' crossover.

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