Multidisciplinary design optimization under uncertainty (MDOUU) is essential for producing robust and reliable aerospace system designs; however, its practical adoption remains limited due to the prohibitive computational cost of solving uncertainty quantification (UQ) problems at each optimization iteration. This work presents a scalable and efficient UQ strategy that enables the solution of large-scale MDOUU problems by combining global sensitivity–based dimension reduction with a shared-grid quadrature evaluation scheme. The proposed approach exploits sparsity in input–output dependence revealed by first-order Sobol indices. Our investigation shows that, in many practical MDOUU problems, each quantity of interest is significantly influenced by only a small subset of uncertain inputs, yielding low-dimensional, output-specific UQ problems. Rather than solving these reduced UQ problems independently, a shared tensor-grid strategy is introduced in which the union of lifted low-dimensional quadrature nodes is evaluated simultaneously. This enables accurate multi-output uncertainty propagation with a computational cost that scales with the maximum reduced dimension rather than the full stochastic dimension. The method is demonstrated on a large-scale MDOUU problem for NASA’s Lift-Plus-Cruise eVTOL configuration, involving 51 design variables, 7 independent uncertain inputs, and 7 coupled constraint quantities. Sensitivity analysis reveals that each quantity of interest depends on at most 2 dominant uncertain inputs. As a result, the proposed method estimates all required statistical moments with errors below 5 percent using only 9 model evaluations, compared to thousands required by Monte Carlo sampling. When embedded within a gradient-based optimization loop, the proposed approach enables the efficient solution of the MDOUU problem and yields designs that are significantly lighter than those obtained using safety-factor-based MDO.