We report the discovery of a large-scale structure at z=3.44 revealed by JWST data in the EGS field. This structure, dubbed "Cosmic Vine", consists of 20 galaxies with spectroscopic redshifts at $3.43<z<3.45$ and six galaxy overdensities with consistent photometric redshifts, making up a vine-like structure extending over a ~4x0.2 pMpc^2 area. The two most massive galaxies (M*~10^10.9 Msun) of the Cosmic Vine are found to be quiescent with bulge-dominated morphologies ($B/T>70\%$). Comparisons with simulations suggest that the Cosmic Vine would form a cluster with halo mass >10^14 Msun 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 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 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.