We illustrate how the two-pole structures of the $\Lambda(1405)$ emerge from the underlying universal chiral dynamics that describe the coupled-channel interactions between octet baryons and pseudo-Nambu-Goldstone bosons. Specifically, we attribute this phenomenon to the form of the leading-order chiral potential, which is of the Weinberg-Tomozawa type. We reveal how the underlying chiral dynamics can be exposed by examining the light-quark mass evolution of the two poles. The latest lattice QCD simulations have indeed found evidence for the existence of the two poles of $\Lambda(1405)$, in qualitative agreement with our predictions. We briefly mention a recent work in which lattice QCD simulations are studied more quantitatively, along with a proposal for how the SU(3) flavor content of the two poles of $\Lambda(1405)$ can be experimentally verified.