We show the existence of quasi-cluster $\mathcal{A}$-structures and cluster Poisson structures on moduli stacks of sheaves with singular support in the alternating strand diagram of grid plabic graphs by studying the microlocal parallel transport of sheaf quantizations of Lagrangian fillings of Legendrian links. The construction is in terms of contact and symplectic topology, showing that there exists an initial seed associated to a canonical relative Lagrangian skeleton. In particular, mutable cluster $\mathcal{A}$-variables are intrinsically characterized via the symplectic topology of Lagrangian fillings in terms of dually $\mathbb{L}$-compressible cycles. New ingredients are introduced throughout this work, including the initial weave associated to a grid plabic graph, cluster mutation along a non-square face of a plabic graph, the concept of the sugar-free hull, and the notion of microlocal merodromy. Finally, a contact geometric realization of the DT-transformation is constructed for shuffle graphs, proving cluster duality for the cluster ensembles.