Quantizations of Poisson algebras over fields of positive characteristic usually carry a large center. On the other hand, quantum D-modules over characteristic p fields, constructed by mod p Gromov-Witten invariants, also carry distinguished central endomorphisms arising from deformations of Steenrod operations. I will explain how to build a bridge between these large center phenomena via duality of symplectic singularities, known as 3D mirror symmetry. This is joint work with Jae Hee Lee.