Quantum Hall states are characterized by a topological invariant, the many-body Chern number, which determines the quantized value of the Hall conductivity. Interestingly, this topological property can also be accessed through a dissipative response, by subjecting the system to a circular drive and comparing excitation rates obtained for opposite orientations of the drive. This quantized circular dichroism assumes that only the bulk contributes to the response. Indeed, in a confined and isolated system, the edge contribution exactly cancels the bulk response. This work explores an important corollary of the latter observation: If properly isolated, the circular dichroic response stemming from the edge of a quantum Hall droplet must be quantized, thus providing an appealing way to probe the many-body Chern number. Importantly, we demonstrate that this quantized edge response is entirely captured by low-energy chiral edge modes, allowing for a universal description of this effect based on Wen's edge theory. Its low-energy nature implies that the quantized edge response can be distinguished from the bulk response in the frequency domain. We illustrate our findings using realistic models of integer and fractional Chern insulators, with different edge geometries, and propose detection schemes suitable for ultracold atoms. Edge dichroic responses emerge as a practical probe for strongly-correlated topological phases, accessible in cold-atom experiments.