Chondrichthyans possess endoskeletal appendages called branchial rays that extend laterally from their hyoid and gill-bearing (branchial) arches. Branchial ray outgrowth, like tetrapod limb outgrowth, is maintained by Sonic hedgehog (Shh) signaling. In limbs, distal endoskeletal elements fail to form in the absence of normal Shh signaling, whereas shortened duration of Shh expression correlates with distal endoskeletal reduction in naturally variable populations. Chondrichthyans also exhibit natural variation with respect to branchial ray distribution--elasmobranchs (sharks and batoids) possess a series of ray-supported septa on their hyoid and gill arches, whereas holocephalans (chimaeras) possess a single hyoid arch ray-supported operculum. Here we show that the elongate hyoid rays of the holocephalan Callorhinchus milii grow in association with sustained Shh expression within an opercular epithelial fold, whereas Shh is only transiently expressed in the gill arches. Coincident with this transient Shh expression, branchial ray outgrowth is initiated in C. milii but is not maintained, yielding previously unrecognized vestigial gill arch branchial rays. This is in contrast to the condition seen in sharks, where sustained Shh expression corresponds to the presence of fully formed branchial rays on the hyoid and gill arches. Considered in light of current hypotheses of chondrichthyan phylogeny, our data suggest that the holocephalan operculum evolved in concert with gill arch appendage reduction by attenuation of Shh-mediated branchial ray outgrowth, and that chondrichthyan branchial rays and tetrapod limbs exhibit parallel developmental mechanisms of evolutionary reduction.