We report the discovery of a large-scale structure at z=3.44 revealed by JWST data in the Extended Groth Strip (EGS) field. This structure, called the Cosmic Vine, consists of 20 galaxies with spectroscopic redshifts at 3.4370\%$). Comparisons with simulations suggest that the Cosmic Vine would form a cluster with halo mass $M_{\rm halo}>10^{14}M_\odot$ at z=0, and the two massive galaxies are likely forming the brightest cluster galaxies (BCGs). The results unambiguously reveal that massive quiescent galaxies can form in growing large-scale structures at z>3, thus disfavoring the environmental quenching mechanisms that require a virialized cluster core. Instead, as suggested by the interacting and bulge-dominated morphologies, the two galaxies are likely quenched by merger-triggered starburst or active galactic nucleus (AGN) feedback before falling into a cluster core. Moreover, we found that the observed specific star formation rates of massive quiescent galaxies in z>3 dense environments are one to two orders of magnitude lower than that of the BCGs in the TNG300 simulation. This discrepancy potentially poses a challenge to the models of massive cluster galaxy formation. Future studies comparing a large sample with dedicated cluster simulations are required to solve the problem.