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 3.43 < z < 3.45 and six galaxy overdensities (4−7σ) with consistent photometric redshifts, making up a vine-like structure extending over a ∼4 × 0.2 pMpc² area. The two most massive galaxies (M_∗ ≈ 10^10.9 M ) 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 M_halo > 10¹⁴M 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.