Mirror symmetry predicts that singular Lagrangian submanifolds with "brane structures" have complex analytic deformation spaces. For conical singularities, the deformation space is closely related to the "augmentation variety" associated to the Chekanov-Eliashberg algebra. I will describe a joint result with Blakey, Chanda and Sun which describes the augmentation varieties for "toric Legendrians" obtained by lifting toric monotone Lagrangians in toric varieties. In particular we prove part of a conjecture of Dimitroglou-Rizell and Golovko relating the augmentation varieties to level sets of the disk superpotential.